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The following three pieces of information completely determine the homeomorphic type of a surface (Massey 1996): 1. Orientability, 2. Number of boundary components, 3. Euler ...

The multiplicative subgroup of all elements in the product of the multiplicative groups k_nu^× whose absolute value is 1 at all but finitely many nu, where k is a number ...

König's line coloring theorem states that the edge chromatic number of any bipartite graph equals its maximum vertex degree. In other words, every bipartite graph is a class ...

A number x such that for all epsilon>0, there exists a member of the set y different from x such that |y-x|<epsilon. The topological definition of limit point P of A is that ...

The unknotting number for a torus knot (p,q) is (p-1)(q-1)/2. This 40-year-old conjecture was proved (Adams 1994) by Kronheimer and Mrowka (1993, 1995).

A mental device used to aid memorization. Common mnemonics for mathematical constants such as e and pi consist of sentences in which the number of letters in each word give ...

A negative matrix is a real or integer matrix (a)_(ij) for which each matrix element is a negative number, i.e., a_(ij)<0 for all i, j. Negative matrices are therefore a ...

A nonpositive matrix is a real or integer matrix (a)_(ij) for which each matrix element is a nonpositive number, i.e., a_(ij)<=0 for all i, j. Nonpositive matrices are ...

The norm topology on a normed space X=(X,||·||_X) is the topology tau consisting of all sets which can be written as a (possibly empty) union of sets of the form B_r(x)={y in ...

Given a convex plane region with area A and perimeter p, A-1/2p<N<=A+1/2p+1, where N is the number of enclosed lattice points (Nosarzewska 1948). This improves on Jarnick's ...