'Beauty is the first test: there is no permanent place in the world for ugly mathematics.' -- G. H. Hardy

'A profound study of nature is the most fertile source of mathematical discoveries.' -- Joseph Fourier

Diagram of special function relationships johndcook.com/blog/2010/09/2…

''Obvious' is the most dangerous word in mathematics.' -- E. T. Bell

Thin plate (biharmonic) splines are an interpolation method with a closed form thanks to the theory of reproducing Hilbert spaces. Defines a Sobolev space of smooth functions. en.wikipedia.org/wiki/Thin_plat… en.wikipedia.org/wiki/Radial_ba… en.wikipedia.org/wiki/Reproduci…

Integrating on R by summing on Q mathoverflow.net/questions/4776…

Sine approximations for small angles johndcook.com/blog/2010/07/2…

'Young men should prove theorems, old men should write books.' -- G. H. Hardy

New post: A couple more variations on an ancient theme johndcook.com/blog/2024/09/0…

Oldies but goldies: S. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilities, 1912. Constructive and quantitative proof of Weierstrass polynomial density in continuous function using Bernstein’s polynomials. en.wikipedia.org/wiki/Bernstein…

'... all exact science is dominated by the concept of approximation.' -- Bertrand Shaw

Einstein summation notation: If an index appears twice in a product, as a subscript and a superscript, sum over that index.

How medieval astronomers made trig tables johndcook.com/blog/2014/11/2…

'The Hilbert transform is, without question, the most important operator in analysis.' -- Steven Krantz