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Consider an n×n (0, 1)-matrix such as [a_(11) a_(23) ; a_(22) a_(34); a_(21) a_(33) ; a_(32) a_(44); a_(31) a_(43) ; a_(42) a_(54); a_(41) a_(53) ; a_(52) a_(64)] (1) for ...

The curve produced by fixed point P on the circumference of a small circle of radius b rolling around the inside of a large circle of radius a>b. A hypocycloid is therefore a ...

Let each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces. Call the resulting polyhedron the local cell. Then the local density is ...

Given a rectangle BCDE, draw EF=DE on an extension of BE. Bisect BF and call the midpoint G. Now draw a semicircle centered at G, and construct the extension of ED which ...

The x-axis is the horizontal axis of a two-dimensional plot in Cartesian coordinates that is conventionally oriented to point to the right (left figure). In three dimensions, ...

Call a projection of a link an almost alternating projection if one crossing change in the projection makes it an alternating projection. Then an almost alternating link is a ...

A d-dimensional framework is a pair (G,p) where G=(V,E) is a graph with vertex set V and edge set E and p:V->R^d is a map that assigns a point in R^d to each vertex of G. The ...

Two totally ordered sets (A,<=) and (B,<=) are order isomorphic iff there is a bijection f from A to B such that for all a_1,a_2 in A, a_1<=a_2 iff f(a_1)<=f(a_2) (Ciesielski ...

Let X be a set. Then a sigma-algebra F is a nonempty collection of subsets of X such that the following hold: 1. X is in F. 2. If A is in F, then so is the complement of A. ...

In nonstandard analysis, the limitation to first-order analysis can be avoided by using a construction known as a superstructure. Superstructures are constructed in the ...