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The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, [2] cloud sizes, [3] the foraging pattern of various species, [4] the sizes of activity patterns of neuronal populations, [5] the frequencies of words in most languages ...
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ l ə ˈ p l ɑː s /), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane).
The Taylor series of any polynomial is the polynomial itself.. The Maclaurin series of 1 / 1 − x is the geometric series + + + +. So, by substituting x for 1 − x, the Taylor series of 1 / x at a = 1 is
The geometric series + + + + … is an infinite series derived from a special type of sequence called a geometric progression, which is defined by just two parameters: the initial term and the common ratio .
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 between an exponential function and its argument .
French curve extrapolation is a method suitable for any distribution that has a tendency to be exponential, but with accelerating or decelerating factors. [3] This method has been used successfully in providing forecast projections of the growth of HIV/AIDS in the UK since 1987 and variant CJD in the UK for a number of years.
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations .
Malthus wrote that all life forms, including humans, have a propensity to exponential population growth when resources are abundant but that actual growth is limited by available resources: "Through the animal and vegetable kingdoms, nature has scattered the seeds of life abroad with the most profuse and liberal hand. ...
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