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A determinant used to determine in which coordinate systems the Helmholtz differential equation is separable (Morse and Feshbach 1953). A determinant S=|Phi_(mn)|=|Phi_(11) ...

A cubic map is three-colorable iff each interior region is bounded by an even number of regions. A non-cubic map bounded by an even number of regions is not necessarily ...

A category modeled after the properties of the category of sets. A category E is a topos if E has finite limits and every object of E has a power object (Barr and Wells 1985, ...

The torsion of a space curve, sometimes also called the "second curvature" (Kreyszig 1991, p. 47), is the rate of change of the curve's osculating plane. The torsion tau is ...

The above two figures are rearrangements of each other, with the corresponding triangles and polyominoes having the same areas. Nevertheless, the bottom figure has an area ...

A subset E of a topological space S is said to be of first category in S if E can be written as the countable union of subsets which are nowhere dense in S, i.e., if E is ...

Let A and B be any sets. Then the product of |A| and |B| is defined as the Cartesian product |A|*|B|=|A×B| (Ciesielski 1997, p. 68; Dauben 1990, p. 173; Moore 1982, p. 37; ...

There exists a system of distinct representatives for a family of sets S_1, S_2, ..., S_m iff the union of any k of these sets contains at least k elements for all k from 1 ...

Given f:X->Y, the image of x is f(x). The preimage of y is then f^(-1)(y)={x|f(x)=y}, or all x whose image is y. Images are elements of the range, while preimages are subsets ...

For any ordinal number alpha, the successor of alpha is alpha union {alpha} (Ciesielski 1997, p. 46). The successor of an ordinal number alpha is therefore the next ordinal, ...