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For K a given knot in S^3, choose a Seifert surface M^2 in S^3 for K and a bicollar M^^×[-1,1] in S^3-K. If x in H_1(M^^) is represented by a 1-cycle in M^^, let x^+ denote ...

A smooth function is a function that has continuous derivatives up to some desired order over some domain. A function can therefore be said to be smooth over a restricted ...

A number taken to the power 2 is said to be squared, so x^2 is called "x squared." This terminology derives from the fact that the area of a square of edge length x is given ...

Taylor's inequality is an estimate result for the value of the remainder term R_n(x) in any n-term finite Taylor series approximation. Indeed, if f is any function which ...

The term cube is used in topology to denote the Cartesian product of any (finite or infinite) number of copies of the closed interval [0,1] equipped with the product topology ...

A transformation x^'=Ax is unimodular if the determinant of the matrix A satisfies det(A)=+/-1. A necessary and sufficient condition that a linear transformation transform a ...

The vertex height of a vertex v in a rooted tree is the number of edges on the longest downward path between v and a tree leaf. The height of the root vertex of a rooted tree ...

The cocktail party graph of order n, also called the hyperoctahedral graph (Biggs 1993, p. 17), n-octahedron graph O_n (Jungerman and Ringel 1978), matching graph (Arvind et ...

Place a point somewhere on a line segment. Now place a second point and number it 2 so that each of the points is in a different half of the line segment. Continue, placing ...

A Smarandache-like function which is defined where S_k(n) is defined as the smallest integer for which n|S_k(n)^k. The Smarandache S_k(n) function can therefore be obtained ...