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Let M be a finitely generated module over a commutative Noetherian ring R. Then there exists a finite set {N_i|1<=i<=l} of submodules of M such that 1. intersection ...

Let V and W be vector spaces over a field F, and let T:V->W be a linear transformation. Assuming the dimension of V is finite, then dim(V)=dim(Ker(T))+dim(Im(T)), where ...

The space of immersions of a manifold in another manifold is homotopically equivalent to the space of bundle injections from the tangent space of the first to the tangent ...

In a network with three graph edges at each graph vertex, the number of Hamiltonian cycles through a specified graph edge is 0 or even.

A special case of Apollonius' problem requiring the determination of a circle touching three mutually tangent circles (also called the kissing circles problem). There are two ...

As originally stated by Gould (1972), GCD{(n-1; k),(n; k-1),(n+1; k+1)} =GCD{(n-1; k-1),(n; k+1),(n+1; k)}, (1) where GCD is the greatest common divisor and (n; k) is a ...

All closed surfaces, despite their seemingly diverse forms, are topologically equivalent to spheres with some number of handles or cross-caps. The traditional proof follows ...

Let P be a matrix of eigenvectors of a given square matrix A and D be a diagonal matrix with the corresponding eigenvalues on the diagonal. Then, as long as P is a square ...

It is always possible to write a sum of sinusoidal functions f(theta)=acostheta+bsintheta (1) as a single sinusoid the form f(theta)=ccos(theta+delta). (2) This can be done ...

For any constructible function f, there exists a function P_f such that for all functions t, the following two statements are equivalent: 1. There exists an algorithm A such ...