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A formula for the generalized Catalan number _pd_(qi). The general formula is (n-q; k-1)=sum_(i=1)^k_pd_(qi)(n-pi; k-i), where (n; k) is a binomial coefficient, although ...

Let V!=(0) be a finite dimensional vector space over the complex numbers, and let A be a linear operator on V. Then V can be expressed as a direct sum of cyclic subspaces.

For 0<=x<=pi/2, 2/pix<=sinx<=x.

The throwing and catching of multiple objects such that at least one is always in the air. Some aspects of juggling turn out to be quite mathematical. The best examples are ...

A point of discontinuity, also called a leap.

A geodesic triangle with oriented boundary yields a curve which is piecewise differentiable. Furthermore, the tangent vector varies continuously at all but the three corner ...

A bistable eight-sided polyhedron discovered by Wunderlich and Schwabe (1986).

A framework is called "just rigid" if it is rigid, but ceases to be so when any single bar is removed. Lamb (1928, pp. 93-94) proved that a necessary (but not sufficient) ...

The Kampyle of Eudoxus is a curve studied by Eudoxus in relation to the classical problem of cube duplication. It is given by the polar equation r=asec^2theta, (1) and the ...

An optical illusion, illustrated above, in which the eye perceives a white upright equilateral triangle where none is actually drawn.