Compressed exponential function
The stretched exponential function

is obtained by inserting a fractional power law into the exponential function. In most applications, it is meaningful only for arguments t between 0 and +∞. With β=1, the usual exponential function is recovered. With a stretching exponent β between 0 and 1, the graph of log f versus t is characteristically stretched, whence the name of the function. The compressed exponential function (with β>1) has less practical importance, with the notable exception of β=2, which gives the normal distribution.
In mathematics, the stretched exponential is also known as the complementary cumulative Weibull distribution. Furthermore, the stretched exponential is the characteristic function (basically the Fourier transform) of the Lévy symmetric alpha-stable distribution.
In physics, the stretched exponential function is often used as a phenomenological description of relaxation in disordered systems. It was first introduced by Rudolf Kohlrausch in 1854 to describe the discharge of a capacitor; therefore it is also called the Kohlrausch function. In 1970, G. Williams and D.C. Watts used the Fourier transform of the stretched exponential to describe dielectric spectra of polymers; in this context, the stretched exponential or its Fourier transform are also called the Kohlrausch-Williams-Watts (KWW) function.
In phenomenological applications, it is often not clear whether the stretched exponential function should apply to the differential or to the integral distribution function—or to neither. In each case one gets the same asymptotic decay, but a different power law prefactor, which makes fits more ambiguous than for simple exponentials. In a few cases it can be shown that the asymptotic decay is a stretched exponential, but the prefactor is usually an unrelated power.

This is an excerpt from the article Compressed exponential function from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
The article Compressed exponential function at en.wikipedia.org was accessed 9 times in the last 30 days. (as of: 05/22/2014)
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Stretched exponential function - Wikipedia, the free encyclopedia
The Compressed exponential function (with β>1) has less practical importance, with the notable exception of β=2, which gives the normal distribution.
en.wikipedia.org/wiki/Stretched_exponential_function
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Compressed Exponential Response Function Arising From a ...
Jan 31, 2013 ... A Compressed exponential function (CEF) is shown to be represented by a distribution of Gaussian functions. The properties and characteristics ...
onlinelibrary.wiley.com/doi/10.1002/macp.201200715/abstract
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Visual Calculus - Transformation of Exponential Functions
Transformations of Exponential Functions ... c - Translate Graph Horizontally; b - Horizontal Stretching or Compression; a - Vertical Stretching or Compression ...
archives.math.utk.edu/visual.calculus/0/shifting.5/
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A relaxation function encompassing the stretched exponential and ...
A simple relaxation function I(t/τ0; α, β) unifying the stretched exponential with the compressed hyperbola is obtained, and its properties studied. The scaling ...
arxiv.org/pdf/0804.2702
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Lesson 6: 2.6 Analyzing Graphs of Exponential Functions
A. Exponential Functions ... B. Applying Graphing Tools-to the Exponential Function y = b" ... Horizontal compression: k > 1 Horizontal expansion: 0 < k
www.wjmouat.com/Teachers/MBidal/PR12_Unit_Notes/Unit2/Lesson7.pdf
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BioMath: Exponential Functions
Graphing Part II: Graphs of Transformed Exponential Functions ... that has been compressed vertically with respect to the base function, f2 (x) = 2 x.
www.biology.arizona.edu/biomath/tutorials/exponential/Graphing2.html
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Transformations explanations on exponential functions and ...
Transformations explanations on exponential functions and logarithmic ... So our function, y = 2^(3x) is a horizontal compression of factor 3 of the parent function ...
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Stretched versus compressed exponential kinetics in α-helix folding
Stretched versus compressed exponential kinetics in α-helix folding ... α-helix with a kinetics that could be fit by a stretched exponential function exp(-(t/τ)β).
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Graphing Exponential Functions
SKETCHING EXPONENTIAL FUNCTIONS ... The graph of the function f(x)=2x. 1 ... compressing it) left and right (imagine stretching the x-axis) by a factor of c.
batty.mullikin.org/shc_courses/mth111/f06/02/exponential.pdf
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Exponential Functions: Introduction
Exponential functions look somewhat similar to functions you have seen before, in that they involve exponents, but there is a big difference, in that the ...
www.purplemath.com/modules/expofcns.htm
Search results for "Compressed exponential function"
Compressed exponential function in science
Compressed Exponential Response Function Arising From a ...
Jan 31, 2013 ... Compressed Exponential Response Function Arising From a Continuous Distribution of Gaussian Decays – Distribution Characteristics.
[PDF]Compressed-Exponential Relaxations in Trehalose Glass1
to compressed-exponential relaxations as the temperature increased. ... Pusan National University. 3. ... Fig.2 The stretched exponent β of trehalose glass with.
Transformation of Exponential Functions - Mathematics Archives
Transformations of Exponential Functions ... c - Translate Graph Horizontally; b - Horizontal Stretching or Compression; a - Vertical Stretching or Compression; d - Translate ... University of Tennessee, Knoxville, Mathematics Department.
Transformation of Exponential Functions - Mathematics Archives
Transformation of Exponential Functions ... c - Translate Graph Horizontally; b - Horizontal Stretching or Compression; a - Vertical Stretching or Compression; d - Translate Graph ... University of Tennessee, Knoxville, Mathematics Department.
Trajectory prediction of saccadic eye movements ... - Journal of Vision
Jul 31, 2013... this misalignment by using a Compressed exponential function to model ... South China Normal University, Guangzhou, Guangdong, China.
Trajectory prediction of saccadic eye movements using a ...
Jul 31, 2013 ... Keywords: saccade, Compressed exponential function, ..... review boards at Schepens Eye Research Institute and Boston University.
BioMath: Exponential Functions - The Biology Project - University of ...
Graphing Part II: Graphs of Transformed Exponential Functions ... that has been compressed vertically with respect to the base function, f2 (x) = 2 x.
Stretched exponential function - Wikipedia, the free encyclopedia
The Compressed exponential function (with β>1) has less practical importance, with the notable exception of β=2, which gives the normal distribution.
Books on the term Compressed exponential function
Anomalous Transport: Foundations and Applications
Rainer Klages, G?nter Radons, Igor M. Sokolov, 2008
11.3 Models of Compressed Exponentials Again, a trivial model leading to both superdiffusion and compressed exponential relaxation (i.e., j3 > 1) is when the noise driving the particle is a long- ranged persistent Gaussian process, in which  ...
Compression: From Cochlea to Cochlear Implants
Sid P. Bacon, Richard R. Fay, Arthur N. Popper, 2004
In electric hearing, loudness growth has been modeled as either a power function or an exponential function (for a review, see Zeng and Shannon 1992). There are generally two methods to obtain the loudness function. One method is to use ...
Photomechanical Actuation of Carbon Nanotubes and Their ...
Shaoxin Lu, 2007
The transient responses in Figure 4.25(a) during “off-on” intervals fit well with a compressed-exponential function: while classical exponential fitting fails to fit the transient data, as shown in Figure 4.26(a). The two fitting parameters are the ...
Information Systems and Data Compression
Jerzy Seidler, 1997
Here the methodology of generalizations is illustrated on a simple but very important in applications example, when the basic functions of a discrete argument are samples of continuous complex exponential functions. EXAMPLE 7.1.1 ...
Still Image and Video Compression with MATLAB
K. S. Thyagarajan, 2011
c01 JWBS049-Thyagarajan September 22, 2010 16:13 Printer Name: Yet to Come 1.3 IMAGE AND VIDEO COMPRESSION TECHNIQUES 7 Further, the residual image has a probability density function, which is a double- sided exponential ...

Blog posts on the term
Compressed exponential function
"A Novel Multi-Exponential Function-based Companding Technique for Unif" by Taleb Moazzeni, Henry Selvaraj et al.
Companding, as a variant of audio level compression, can help reduce the dynamic range of an audio signal. In analog (digital) systems, this can increase the signal-to-noise ratio (signal to quantization noise ratio) achieved during transmission. The μ-law algorithm that is primarily used in the digital telecommunication systems of North America and Japan, adapts a companding scheme that can expand small signals and compress large signals especially at the presence of high peak signals. In this paper, we present a novel multi-exponential companding function that can achieve more uniform compression on both large and small signals so that the relative signal strength over the time is preserved. That is, although larger signals may get considerably compressed, unlike μ-law algorithm, it is guaranteed that these signals after companding will definitely not be smaller than expanded signals that were originally small. Performance of the proposed algorithm is compared with μ-law using real audio signal, and results show that the proposed companding algorithm can achieve much smaller quantization errors with a modest increase in computation time.
digitalscholarship.unlv.edu/ece_fac_articles/275/
Algorithms | Free Full-Text | Laplace–Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source Implementation “libkww”
The C library libkww provides functions to compute the Kohlrausch–Williams– Watts function, i.e., the Laplace–Fourier transform of the stretched (or compressed) exponential function exp(-tβ ) for exponents β between 0.1 and 1.9 with double precision. Analytic error bounds are derived for the low and high frequency series expansions. For intermediate frequencies, the numeric integration is enormously accelerated by using the Ooura–Mori double exponential transformation. The primitive of the cosine transform needed for the convolution integrals is also implemented. The software is hosted at http://apps.jcns.fz-juelich.de/kww; version 3.0 is deposited as supplementary material to this article.
www.mdpi.com/1999-4893/5/4/604
CraigsMaths » Blog Archive » Graphs of exponential functions
By comparing exponential and logarithmic functions with their simplest forms it is easy to sketch their graphs using a transformational approach. All graphs in this article were created with OpenPlaG.
www.craigsmaths.com/number/graphs-of-exponential-functions/
Learning Physics, Metaphors, and Quantum Fields | Theory and Practice of Trying to Combine Just Anything
In my series on Quantum Field Theory I wanted to document my own learning endeavors but it has turned into a meta-contemplation on the 'explain-ability' of theoretical physics. Initially I had been motivated by a comment David Tong made in his introductory lecture: Comparing different QFT books he states that Steven Weinberg's books are hard reads because…
elkement.wordpress.com/2013/11/04/learning-physics-metaphors-and-quantum-fields/
Shtetl-Optimized » Blog Archive » A not-quite-exponential dilemma
A commenter on my last post writes: Scott, it’s your blog – but can’t we switch back to some QC or TCS topics? I confess: after three years of staid, dignified posts about quantum computing and theoretical computer science, I somehow did the unthinkable, and let this once-respected blog become less about elucidating research than procrastinating from it. Many readers, or so they tell me, rely on Shtetl-Optimized to keep abreast of the latest theoretical insights.
www.scottaaronson.com/blog/?p=394
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