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Given a commutative ring R, an R-algebra H is a Hopf algebra if it has additional structure given by R-algebra homomorphisms Delta:H->H tensor _RH (1) (comultiplication) and ...

In the early 1950s, Ernst Straus asked 1. Is every region illuminable from every point in the region? 2. Is every region illuminable from at least one point in the region? ...

The isotomic conjugate of a point is the point of concurrence Q of the isotomic lines relative to a point P. The isotomic conjugate alpha^':beta^':gamma^' of a point with ...

The Kakeya needle problems asks for the plane figure of least area in which a line segment of width 1 can be freely rotated (where translation of the segment is also ...

A 4-state two-dimensional Turing machine invented in the 1980s. The ant starts out on a grid containing black and white cells, and then follows the following set of rules. 1. ...

An edge-magic graph is a labeled graph with e graph edges labeled with distinct elements {1,2,...,e} so that the sum of the graph edge labels at each graph vertex is the ...

In the mice problem, also called the beetle problem, n mice start at the corners of a regular n-gon of unit side length, each heading towards its closest neighboring mouse in ...

The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is ...

If equilateral triangles DeltaABE_(AB), DeltaBCE_(BC), and DeltaACE_(AC) are erected externally on the sides of any triangle DeltaABC, then their centers N_(AB), N_(BC), and ...

The Neuberg cubic Z(X_(30)) of a triangle DeltaABC is the locus of all points P whose reflections in the sidelines BC, CA, and ABform a triangle perspective to DeltaABC. It ...