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Let M^n be an n-manifold and let F={F_alpha} denote a partition of M^n into disjoint pathwise-connected subsets. Then F is called a foliation of M^n of codimension c (with ...

A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form alpha in the open unit ball in R^n. Since alpha is a top-dimensional ...

A general quartic surface defined by x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c=0 (1) (Gray 1997, p. 314). The above two images correspond to (a,b,c)=(0,0,-1), and ...

The granny knot is a composite knot of six crossings consisting of a knot sum of two left-handed trefoils with the same orientation (Rolfsen 1976, p. 220). The granny knot ...

A Hajós group is a group for which all factorizations of the form (say) Z_n=A direct sum B have A or B periodic, where the period is a divisor of n. Hajós groups arose after ...

A linkage which draws the inverse of a given curve. It can also convert circular to linear motion. The rods satisfy AB=CD and BC=DA, and O, P, and P^' remain collinear while ...

The construction of polyhedra using identical building blocks. The illustrations above show such constructions for the cuboctahedron, octahedron rhombic dodecahedron, and ...

A fair coin is tossed an even 2n number of times. Let D=|H-T| be the absolute difference in the number of heads and tails obtained. Then the probability distribution is given ...

An abstract algebra concerned with results valid for many different kinds of spaces. Modules are the basic tools used in homological algebra.

An identity is a mathematical relationship equating one quantity to another (which may initially appear to be different).