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The series sumf(n) for a monotonic nonincreasing f(x) is convergent if lim_(x->infty)^_(e^xf(e^x))/(f(x))<1 and divergent if lim_(x->infty)__(e^xf(e^x))/(f(x))>1.

The partial differential equation R[u](u_(rr)+(u_r)/r+u_(zz))=u_r^2+u_z^2, where R[u] is the real part of u (Calogero and Degasperis 1982, p. 62; Zwillinger 1997, p. 131).

A closed ideal I in a C^*-algebra A is called essential if I has nonzero intersection with every other nonzero closed ideal A or, equivalently, if aI={0} implies a=0 for all ...

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

A bounded operator U on a Hilbert space H is called essentially unitary if U^*U-I and UU^*-I are compact operators.

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 three-dimensional shadow of a four-dimensional Klein bottle.

The paradox "This statement is false," stated in the fourth century BC. It is a sharper version of the Epimenides paradox, "All Cretans are liars...One of their own poets has ...

A more common way to describe a Euclidean ring.

A weighted graph in which the weights are equal to the Euclidean lengths of the edges in a specified embedding (Skiena 1990, pp. 201 and 252).