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When =, the differential equation is linear.When =, it is separable.In these cases, standard techniques for solving equations of those forms can be applied. For and , the substitution = reduces any Bernoulli equation to a linear differential equation
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables X i are identically distributed and independent. Prosaically, a Bernoulli ...
Bernoulli's principle is a key concept in fluid dynamics that relates pressure, density, speed and height. Bernoulli's principle states that an increase in the speed of a parcel of fluid occurs simultaneously with a decrease in either the pressure or the height above a datum. [1]:
Being inviscid and irrotational, Bernoulli's equation allows the solution for the pressure field to be obtained directly from the velocity field: = +, where the constants U and p ∞ appear so that p → p ∞ far from the cylinder, where V = U. Using V 2 = V 2 r + V 2
The Bernoulli equation applicable to incompressible flow shows that the stagnation pressure is equal to the dynamic pressure and static pressure combined. [ 1 ] : § 3.5 In compressible flows , stagnation pressure is also equal to total pressure as well, provided that the fluid entering the stagnation point is brought to rest isentropically .
Entropy of a Bernoulli trial (in shannons) as a function of binary outcome probability, called the binary entropy function.. In information theory, the binary entropy function, denoted or (), is defined as the entropy of a Bernoulli process (i.i.d. binary variable) with probability of one of two values, and is given by the formula:
Eq.2b is a fundamental equation for most of discrete models. The equation can be solved by recurrence and iteration method for a manifold. It is clear that Eq.2a is limiting case of Eq.2b when ∆X → 0. Eq.2a is simplified to Eq.1 Bernoulli equation without the potential energy term when β=1 whilst Eq.2 is simplified to Kee's model [6] when β=0
The Bernoulli scheme, as any stochastic process, may be viewed as a dynamical system by endowing it with the shift operator T where T ( x k ) = x k + 1 . {\displaystyle T(x_{k})=x_{k+1}.} Since the outcomes are independent, the shift preserves the measure, and thus T is a measure-preserving transformation .