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The socle of a group G is the subgroup generated by its minimal normal subgroups. For example, the symmetric group S_4 has two nontrivial normal subgroups: A_4 and ...

For a graph G and a subset S of the vertex set V(G), denote by N_G[S] the set of vertices in G which are in S or adjacent to a vertex in S. If N_G[S]=V(G), then S is said to ...

The problem in calculus of variations to find the minimal surface of a boundary with specified constraints (usually having no singularities on the surface). In general, there ...

A cylindric section is the intersection of a plane with a right circular cylinder. It is a circle (if the plane is at a right angle to the axis), an ellipse, or, if the plane ...

The elastica formed by bent rods and considered in physics can be generalized to curves in a Riemannian manifold which are a critical point for ...

Exponential growth is the increase in a quantity N according to the law N(t)=N_0e^(lambdat) (1) for a parameter t and constant lambda (the analog of the decay constant), ...

Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to ...

A generalization to a quartic three-dimensional surface is the quartic surface of revolution (x^4-ax^3)+a^2(y^2+z^2)=0, (1) illustrated above. With a=1, this surface is ...

A polar zonohedron is a convex zonohedron derived from the star which joins opposite vertices of any right n-gonal prism (for n even) or antiprism (for n odd). The faces of ...

A curve alpha on a regular surface M is a principal curve iff the velocity alpha^' always points in a principal direction, i.e., S(alpha^')=kappa_ialpha^', where S is the ...