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The conjecture proposed by Catalan in 1888 and extended by E. Dickson that each aliquot sequence ends in a prime, a perfect number, or a set of sociable numbers. The ...

A dilation that is not merely a translation. Two triangles related by a central dilation are said to be perspective triangles because the lines joining corresponding vertices ...

Characteristic classes are cohomology classes in the base space of a vector bundle, defined through obstruction theory, which are (perhaps partial) obstructions to the ...

The space of continuously differentiable functions is denoted C^1, and corresponds to the k=1 case of a C-k function.

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.

A proof which can be accomplished using only real numbers (i.e., real analysis instead of complex analysis; Hoffman 1998, pp. 92-93).

The radical circle of the excircles has center at the Spieker center X_(10) and radius R_E=1/2sqrt((a^2b+ab^2+a^2c+abc+b^2c+ac^2+bc^2)/(a+b+c)). Its circle function is ...

A fixed point is a point that does not change upon application of a map, system of differential equations, etc. In particular, a fixed point of a function f(x) is a point x_0 ...

An expression occurring in existential sentences. "For some x" is the same as " exists x." Unlike in everyday language, it is does not necessarily refer to a plurality of ...

The Heisenberg group H^n in n complex variables is the group of all (z,t) with z in C^n and t in R having multiplication (w,t)(z,t^')=(w+z,t+t^'+I[w^*z]) (1) where w^* is the ...