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A smooth structure on a topological manifold (also called a differentiable structure) is given by a smooth atlas of coordinate charts, i.e., the transition functions between ...

The space groups in two dimensions are called wallpaper groups. In three dimensions, the space groups are the symmetry groups possible in a crystal lattice with the ...

The square torus is the quotient of the plane by the integer lattice.

The Stieltjes integral is a generalization of the Riemann integral. Let f(x) and alpha(x) be real-valued bounded functions defined on a closed interval [a,b]. Take a ...

Stochastic is synonymous with "random." The word is of Greek origin and means "pertaining to chance" (Parzen 1962, p. 7). It is used to indicate that a particular subject is ...

The first strong law of small numbers (Gardner 1980, Guy 1988, 1990) states "There aren't enough small numbers to meet the many demands made of them." The second strong law ...

A number n such that sigma^2(n)=sigma(sigma(n))=2n, where sigma(n) is the divisor function is called a superperfect number. Even superperfect numbers are just 2^(p-1), where ...

Surreal numbers are the most natural collection of numbers which includes both the real numbers and the infinite ordinal numbers of Georg Cantor. They were invented by John ...

In the early 1960s, B. Birch and H. P. F. Swinnerton-Dyer conjectured that if a given elliptic curve has an infinite number of solutions, then the associated L-series has ...

Ramanujan's Dirichlet L-series is defined as f(s)=sum_(n=1)^infty(tau(n))/(n^s), (1) where tau(n) is the tau function. Note that the notation F(s) is sometimes used instead ...