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Let S_n be the set of permutations of {1, 2, ..., n}, and let sigma_t be the continuous time random walk on S_n that results when randomly chosen transpositions are performed ...

Buffon's needle problem asks to find the probability that a needle of length l will land on a line, given a floor with equally spaced parallel lines a distance d apart. The ...

Given any triangle ABC, the signed sum of perpendicular distances from the circumcenter O to the sides (i.e., signed lengths of the pedal lines from O) is OO_A+OO_B+OO_C=R+r, ...

The quartic surface obtained by replacing the constant b in the equation of the Cassini ovals with b=z, obtaining [(x-a)^2+y^2][(x+a)^2+y^2]=z^4. (1) As can be seen by ...

If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the ...

Cluster analysis is a technique used for classification of data in which data elements are partitioned into groups called clusters that represent collections of data elements ...

The cocktail party graph of order n, also called the hyperoctahedral graph (Biggs 1993, p. 17), n-octahedron graph O_n (Jungerman and Ringel 1978), matching graph (Arvind et ...

A metric space X which is not complete has a Cauchy sequence which does not converge. The completion of X is obtained by adding the limits to the Cauchy sequences. For ...

Given two crossed ladders resting against two buildings, what is the distance between the buildings? Let the height at which they cross be h and the lengths of the ladders ...

A two-person game where each of the players can take as long as he likes to prepare his move, but the other can take advantage of his hesitation. In this conflict situation, ...