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The solutions to some of these problems become well known and are the exemplars of the field. [26] Those who study a scientific discipline are expected to know its exemplars. There is no fixed set of exemplars, but for a physicist today it would probably include the harmonic oscillator from mechanics and the hydrogen atom from quantum mechanics ...
Exemplar, a well-known science problem and its solution, from Thomas Kuhn's The Structure of Scientific Revolutions; Exemplar, the first name for the ship USS Dorothea L. Dix; Exemplar, in exemplification theory, an illustrative representation of information or an event
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In the classical central-force problem of classical mechanics, some potential energy functions () produce motions or orbits that can be expressed in terms of well-known functions, such as the trigonometric functions and elliptic functions. This article describes these functions and the corresponding solutions for the orbits.
One important drawback for applications of the solution of the classical secretary problem is that the number of applicants must be known in advance, which is rarely the case. One way to overcome this problem is to suppose that the number of applicants is a random variable N {\displaystyle N} with a known distribution of P ( N = k ) k = 1 , 2 ...
Problems 1, 2, 5, 6, [a] 9, 11, 12, 15, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems. That leaves 8 (the Riemann hypothesis), 13 and 16 [b] unresolved. Problems 4 and 23 are considered as too vague to ever be described as solved; the withdrawn 24 would also be in ...
Noteworthy examples of vacuum solutions, electrovacuum solutions, and so forth, are listed in specialized articles (see below). These solutions contain at most one contribution to the energy–momentum tensor, due to a specific kind of matter or field. However, there are some notable exact solutions which contain two or three contributions ...
The solution (,), as a function of for a fixed time >, is generally smoother than the initial data (,) = (), according to parabolic regularity theory. For a nonlinear parabolic PDE, a solution of an initial/boundary-value problem might explode in a singularity within a finite amount of time.