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Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of slopes of soil- and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. It is performed to assess the safe design of a human-made or natural slopes (e.g. embankments, road cuts, open-pit mining ...
In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the statistical limits to possible data compression for data whose source is an independent identically-distributed random variable, and the operational meaning of the Shannon entropy . Named after Claude Shannon, the source coding theorem shows ...
An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies". Another example: f(x 1, ..., x n) = const · exp(−(| x 1 | α + ⋯ + | x n | α) β) where α ...
For example, the time (or the number of steps) it takes to complete a problem of size n might be found to be T(n) = 4n 2 − 2n + 2. As n grows large, the n 2 term will come to dominate, so that all other terms can be neglected—for instance when n = 500, the term 4n 2 is 1000 times as large as the 2n term. Ignoring the latter would have ...
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted ) occurs. [ 2] For example, we can define rolling a 6 on some ...
For a different example, in decision theory, an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their utility function. It is possible to construct an expected value equal to the probability of an event by taking the expectation of an indicator function that is one if ...
The Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample rate must be at least twice the bandwidth of the signal to avoid aliasing.
Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named after James Stirling, though a related but less precise result was first stated ...