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As with the Hilbert problems, one of the prize problems (the Poincaré conjecture) was solved relatively soon after the problems were announced. The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise.
A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function f. Hilbert space methods provide one possible answer to this question. [46] The functions e n (θ) = e 2πinθ form an orthogonal basis of the Hilbert space L 2 ([0, 1]).
The space of all test functions, ... The justification for this common practice is detailed below. ... is even a Hilbert space. [7]
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein , Israel Gohberg and others.
This problem is more commonly called the Riemann–Hilbert problem.It led to several bijective correspondences known as 'Riemann–Hilbert correspondences', for flat algebraic connections with regular singularities and more generally regular holonomic D-modules or flat algebraic connections with regular singularities on principal G-bundles, in all dimensions.
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit ...
The second problem also remains unsolved: no upper bound for the number of limit cycles is known for any n > 1, and this is what usually is meant by Hilbert's sixteenth problem in the field of dynamical systems. The Spanish Royal Society for Mathematics published an explanation of Hilbert's sixteenth problem. [2]
The particular case of n = 2 was already solved by Hilbert in 1893. [5] The general problem was solved in the affirmative, in 1927, by Emil Artin, [6] for positive semidefinite functions over the reals or more generally real-closed fields. An algorithmic solution was found by Charles Delzell in 1984. [7]