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This type of differential equation describes exponential growth or exponential decay, depending on the sign of the constant , which is just the expected number of neutrons after one average neutron lifetime has elapsed:
This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case. Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering.
Rate equation. In chemistry, the rate equation (also known as the rate law or empirical differential rate equation) is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters (normally rate coefficients and partial orders of reaction) only ...
Empirical examples. 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, cloud sizes, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages ...
By the asymptotic formula, the probability that empirical distribution ^ deviates from the actual distribution decays exponentially, at a rate (^ ‖). The more experiments and the more different p ^ {\displaystyle {\hat {p}}} is from p {\displaystyle p} , the less likely it is to see such an empirical distribution.
Lutetium–hafnium dating is a geochronological dating method utilizing the radioactive decay system of lutetium –176 to hafnium –176. [1] With a commonly accepted half-life of 37.1 billion years, [1] [2] the long-living Lu–Hf decay pair survives through geological time scales, thus is useful in geological studies. [1]
The exponential growth rate of the matrix powers A k as k → ∞ is controlled by the eigenvalue of A with the largest absolute value . The Perron–Frobenius theorem describes the properties of the leading eigenvalue and of the corresponding eigenvectors when A is a non-negative real square matrix.
Decay correction is one way of working out what the radioactivity would have been at the time it was taken, rather than at the time it was tested. For example, the isotope copper-64, commonly used in medical research, has a half-life of 12.7 hours. If you inject a large group of animals at "time zero", but measure the radioactivity in their ...