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For n>=1, let u and v be integers with u>v>0 such that the Euclidean algorithm applied to u and v requires exactly n division steps and such that u is as small as possible ...

A zonohedron which is the dual of the dodecadodecahedron U_(36) and Wenninger dual W_(73). The medial rhombic triacontahedron contains interior pentagrammic vertices which ...

What is the longest ladder that can be moved around a right-angled hallway of unit width? For a straight, rigid ladder, the answer is 2sqrt(2), which allows the ladder to ...

An odious number is a nonnegative number that has an odd number of 1s in its binary expansion. The first few odious numbers are therefore 1, 2, 4, 7, 8, 11, 13, 14, 16, 19, ...

The theory of natural numbers defined by the five Peano's axioms. Paris and Harrington (1977) gave the first "natural" example of a statement which is true for the integers ...

Any entire analytic function whose range omits two points must be a constant function. Of course, an entire function that omits a single point from its range need not be a ...

In its simplest form, the principle of permanence states that, given any analytic function f(z) defined on an open (and connected) set U of the complex numbers C, and a ...

Quantization is a nonlinear process which generates additional frequency components (Thompson et al. 1986). This means that the signal is no longer band-limited, so the ...

Following Ramanujan (1913-1914), write product_(k=1,3,5,...)^infty(1+e^(-kpisqrt(n)))=2^(1/4)e^(-pisqrt(n)/24)G_n (1) ...

The second Morley triangle is made by rotating line BC toward vertex A about vertex B by angle (B+2pi)/3. It is an equilateral triangle. It has trilinear vertex matrix [1 ...