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The direct method may often be applied with success when the space is a subset of a separable reflexive Banach space. In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence ( u n ) {\displaystyle (u_{n})} in V {\displaystyle V} has a subsequence that converges to some u 0 {\displaystyle u_{0}} in W {\displaystyle ...
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.
Hilbert noted that there existed methods for solving partial differential equations where the function's values were given at the boundary, but the problem asked for methods for solving partial differential equations with more complicated conditions on the boundary (e.g., involving derivatives of the function), or for solving calculus of variation problems in more than 1 dimension (for example ...
For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because ...
A common example of an NP problem not known to be in P is the Boolean satisfiability problem. Most mathematicians and computer scientists expect that P ≠ NP; however, it remains unproven. [16] The official statement of the problem was given by Stephen Cook. [17]
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 .
It is also called the constant of variation or constant of proportionality. Given such a constant k , the proportionality relation ∝ with proportionality constant k between two sets A and B is the equivalence relation defined by { ( a , b ) ∈ A × B : a = k b } . {\displaystyle \{(a,b)\in A\times B:a=kb\}.}