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Let f be a real-valued, continuous, and strictly increasing function on [0,c] with c>0. If f(0)=0, a in [0,c], and b in [0,f(c)], then int_0^af(x)dx+int_0^bf^(-1)(x)dx>=ab, ...

The gyroid, illustrated above, is an infinitely connected periodic minimal surface containing no straight lines (Osserman 1986) that was discovered by Schoen (1970). ...

Gelfond's theorem, also called the Gelfond-Schneider theorem, states that a^b is transcendental if 1. a is algebraic !=0,1 and 2. b is algebraic and irrational. This provides ...

Arrange copies of the n digits 1, ..., n such that there is one digit between the 1s, two digits between the 2s, etc. For example, the unique (modulo reversal) n=3 solution ...

The Nagel line is the term proposed for the first time in this work for the line on which the incenter I, triangle centroid G, Spieker center Sp, and Nagel point Na lie. ...

If a matrix A has a matrix of eigenvectors P that is not invertible (for example, the matrix [1 1; 0 1] has the noninvertible system of eigenvectors [1 0; 0 0]), then A does ...

The thin plate spline is the two-dimensional analog of the cubic spline in one dimension. It is the fundamental solution to the biharmonic equation, and has the form ...

Just as the ratio of the arc length of a semicircle to its radius is always pi, the ratio P of the arc length of the parabolic segment formed by the latus rectum of any ...

For every positive integer n, there exists a circle which contains exactly n lattice points in its interior. H. Steinhaus proved that for every positive integer n, there ...

A Fibonacci prime is a Fibonacci number F_n that is also a prime number. Every F_n that is prime must have a prime index n, with the exception of F_4=3. However, the converse ...