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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 ...

Let E(k) and K(k) be complete elliptic integrals of the first and second kinds, with E^'(k) and K^'(k) the complementary integrals. Then ...

Linkages are linked (or constrained) mechanical structures such as those used in robots, excavators, engines, etc. They are also extensively used for character animation in ...

The longest path problem asks to find a path of maximum length in a given graph. The problem is NP-complete, but there exists an efficient dynamic programming solution for ...

The map-Airy distribution is a statistical distribution having probability density function and distribution function P(x) = 2e^(-2x^3/3)[xAi(x^2)-Ai^'(x^2)] (1) D(x) = (2) ...

When a pair of non-incident edges of a tetrahedron is chosen, the midpoints of the remaining 4 edges are the vertices of a planar parallelogram. Furthermore, the area of this ...

An expanded paper bag, cushion, or pillow has a distinctive shape. Given the dimensions of a flat rectangular bag with dimensions a, b, bounds for the maximum volume of ...

Let p={a_1,a_2,...,a_n} be a permutation. Then i is a permutation ascent if a_i<a_(i+1). For example, the permutation {1,2,3,4} is composed of three ascents, namely {1,2}, ...

The index of a permutation p is defined as the sum of all subscripts j such that p_j>p_(j+1), for 1<=j<=n. MacMahon (1960) proved that the number of permutations of size n ...

"Q.E.D." (sometimes written "QED") is an abbreviation for the Latin phrase "quod erat demonstrandum" ("that which was to be demonstrated"), a notation which is often placed ...