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Differential Geometry

The term Borel hierarchy is used to describe a collection of subsets of R defined inductively as follows: Level one consists of all open and closed subsets of R, and upon ...

A subset F subset R of the real numbers is said to be an F_sigma set provided F is the countable union of closed sets. The name F_sigma comes from French: The F stands for ...

If {f_n} is a sequence of nonnegative measurable functions, then intlim inf_(n->infty)f_ndmu<=lim inf_(n->infty)intf_ndmu. (1) An example of a sequence of functions for which ...

A subset G subset R of the real numbers is said to be a G_delta set provided G is the countable intersection of open sets. The name G_delta comes from German: The G stands ...

The Littlewood conjecture states that for any two real numbers x,y in R, lim inf_(n->infty)n|nx-nint(nx)||ny-nint(ny)|=0 where nint(z) denotes the nearest integer function. ...

The set of L^p-functions (where p>=1) generalizes L2-space. Instead of square integrable, the measurable function f must be p-integrable for f to be in L^p. On a measure ...

Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are ...

For an integer n>=2, let gpf(x) denote the greatest prime factor of n, i.e., the number p_k in the factorization n=p_1^(a_1)...p_k^(a_k), with p_i<p_j for i<j. For n=2, 3, ...

A sequence s_n(x) is called a Sheffer sequence iff its generating function has the form sum_(k=0)^infty(s_k(x))/(k!)t^k=A(t)e^(xB(t)), (1) where A(t) = A_0+A_1t+A_2t^2+... ...