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In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum ( functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf .
The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, [2] which was the first problem in the field to be formulated and correctly solved, [2] and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to ...
f: I → R is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property. This statement is also known as the Banach-Zareckiǐ theorem. [8] If f: I → R is absolutely continuous and g: R → R is globally Lipschitz-continuous, then the composition g ∘ f is absolutely continuous.
Given two metric spaces (X, d X) and (Y, d Y), where d X denotes the metric on the set X and d Y is the metric on set Y, a function f : X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x 1 and x 2 in X,
The immutability of these fundamental constants is an important cornerstone of the laws of physics as currently known; the postulate of the time-independence of physical laws is tied to that of the conservation of energy (Noether's theorem), so that the discovery of any variation would imply the discovery of a previously unknown law of force. [3]
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that ...
In the case of Brownian motion on , the choice of constants =, =, = (+) will work in the Kolmogorov continuity theorem. Moreover, for any positive integer m {\displaystyle m} , the constants α = 2 m {\displaystyle \alpha =2m} , β = m − 1 {\displaystyle \beta =m-1} will work, for some positive value of K {\displaystyle K} that depends on n ...