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An equation of the form f(x)=b (mod m), (1) where the values of 0<=x<m for which the equation holds are sought. Such an equation may have none, one, or many solutions. There ...

Let P be a prime ideal in D_m not containing m. Then (Phi(P))=P^(sumtsigma_t^(-1)), where the sum is over all 1<=t<m which are relatively prime to m. Here D_m is the ring of ...

Differential Geometry

A dimension also called the fractal dimension, Hausdorff dimension, and Hausdorff-Besicovitch dimension in which nonintegral values are permitted. Objects whose capacity ...

The number of ways to arrange n distinct objects along a fixed (i.e., cannot be picked up out of the plane and turned over) circle is P_n=(n-1)!. The number is (n-1)! instead ...

The average number of regions into which n randomly chosen planes divide a cube is N^_(n)=1/(324)(2n+23)n(n-1)pi+n+1 (Finch 2003, p. 482). The maximum number of regions is ...

The maximum number of pieces into which a cylinder can be divided by n oblique cuts is given by f(n) = (n+1; 3)+n+1 (1) = 1/6(n+1)(n^2-n+6) (2) = 1/6(n^3+5n+6), (3) where (a; ...

A minimum edge cover is an edge cover having the smallest possible number of edges for a given graph. The size of a minimum edge cover of a graph is known as the edge cover ...

For the rational curve of an unperturbed system with rotation number r/s under a map T (for which every point is a fixed point of T^s), only an even number of fixed points ...

The twinplex graph is the graph illustrated above in a number of embeddings which corresponds to the graph labeled Gamma_2 in Fischer and Little (2011). It is a cubic ...