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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. [1]
Any Riemann sum of f on [0, 1] will have the value 1, therefore the Riemann integral of f on [0, 1] is 1. Let : [,] be the indicator function of the rational numbers in [0, 1]; that is, takes the value 1 on rational numbers and 0 on irrational numbers. This function does not have a Riemann integral.
Riemann mapping theorem (complex analysis) Riemann series theorem (mathematical series) Riemann's existence theorem (algebraic geometry) Riemann's theorem on removable singularities (complex analysis) Riemann–Roch theorem (Riemann surfaces, algebraic curves) Riemann–Roch theorem for smooth manifolds (differential topology)
In mathematics, the Riemann–Liouville integral associates with a real function: another function I α f of the same kind for each value of the parameter α > 0.The integral is a manner of generalization of the repeated antiderivative of f in the sense that for positive integer values of α, I α f is an iterated antiderivative of f of order α.
The Riemann–Stieltjes integral admits integration by parts in the form () = () () ()and the existence of either integral implies the existence of the other. [2]On the other hand, a classical result [3] shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with α + β > 1 .
However, the Riemann–Lebesgue lemma does not hold for arbitrary distributions. For example, the Dirac delta function distribution formally has a finite integral over the real line, but its Fourier transform is a constant and does not vanish at infinity.
Suppose that X is a smooth complex algebraic variety.. Riemann–Hilbert correspondence (for regular singular connections): there is a functor Sol called the local solutions functor, that is an equivalence from the category of flat connections on algebraic vector bundles on X with regular singularities to the category of local systems of finite-dimensional complex vector spaces on X.
Generally speaking, Riemann solvers are specific methods for computing the numerical flux across a discontinuity in the Riemann problem. [1] They form an important part of high-resolution schemes; typically the right and left states for the Riemann problem are calculated using some form of nonlinear reconstruction, such as a flux limiter or a WENO method, and then used as the input for the ...