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The study of definable sets and functions in polish spaces.

A test for the convergence of Fourier series. Let phi_x(t)=f(x+t)+f(x-t)-2f(x), then if int_0^pi(|phi_x(t)|dt)/t is finite, the Fourier series converges to f(x) at x.

If (f,U) and (g,V) are functions elements, then (g,V) is a direct analytic continuation of (f,U) if U intersection V!=emptyset and f and g are equal on U intersection V.

Given any real number theta and any positive integer N, there exist integers h and k with 0<k<=N such that |ktheta-h|<1/N. A slightly weaker form of the theorem states that ...

Let (X,B,mu) be a measure space and let E be a measurable set with mu(E)<infty. Let {f_n} be a sequence of measurable functions on E such that each f_n is finite almost ...

Ellipsoidal calculus is a method for solving problems in control and estimation theory having unknown but bounded errors in terms of sets of approximating ellipsoidal-value ...

A singular point a for which f(z)(z-a)^n is not differentiable for any integer n>0.

A topology arising from a sheaf of continuous functions. It derives a natural topology from the projection operator. Etale spaces are examples of space that are not T2.

A distribution which arises in the study of half-integer spin particles in physics, P(k)=(k^s)/(e^(k-mu)+1). (1) Its integral is given by int_0^infty(k^sdk)/(e^(k-mu)+1) = ...

The ring of fractions of an integral domain. The field of fractions of the ring of integers Z is the rational field Q, and the field of fractions of the polynomial ring ...