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For a braid with M strands, R components, P positive crossings, and N negative crossings, {P-N<=U_++M-R if P>=N; P-N<=U_-+M-R if P<=N, (1) where U_+/- are the smallest number ...

Spherical mirrors were a popular subject for M. C. Escher's lithographs, including "Still Life with a Spherical Mirror" (Bool et al. 1982, p. 261; Forty 2003, Plate 23), ...

The tau conjecture, also known as Ramanujan's hypothesis after its proposer, states that tau(n)∼O(n^(11/2+epsilon)), where tau(n) is the tau function. This was proven by ...

A Tauberian theorem is a theorem that deduces the convergence of an series on the basis of the properties of the function it defines and any kind of auxiliary hypothesis ...

A square matrix with nonzero elements only on the diagonal and slots horizontally or vertically adjacent the diagonal (i.e., along the subdiagonal and superdiagonal), [a_(11) ...

Geometry

A convex polyhedron can be defined algebraically as the set of solutions to a system of linear inequalities mx<=b, where m is a real s×3 matrix and b is a real s-vector. ...

A cyclide is a pair of focal conics which are the envelopes of two one-parameter families of spheres, sometimes also called a cyclid. The cyclide is a quartic surface, and ...

There are a number of mathematical curves that produced heart shapes, some of which are illustrated above. A "zeroth" curve is a rotated cardioid (whose name means ...

A Heronian tetrahedron, also called a perfect tetrahedron, is a (not necessarily regular) tetrahedron whose sides, face areas, and volume are all rational numbers. It ...