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Let suma_k and sumb_k be two series with positive terms and suppose lim_(k->infty)(a_k)/(b_k)=rho. If rho is finite and rho>0, then the two series both converge or diverge.

If algebraic integers alpha_1, ..., alpha_n are linearly independent over Q, then e^(alpha_1), ..., e^(alpha_n) are algebraically independent over Q. The ...

A member of the smallest algebraically closed subfield L of C which is closed under the exponentiation and logarithm operations.

A branch point whose neighborhood of values wrap around an infinite number of times as their complex arguments are varied. The point z=0 under the function lnz is therefore a ...

A polynomial is called logarithmically concave (or log-concave) if the sequence of its coefficients is logarithmically concave. If P(x) is log-convex and Q(x) is unimodal, ...

If a sphere is covered by three closed sets, then one of them must contain a pair of antipodal points.

A number triangle of order n with entries 1 to n such that entries are nondecreasing across rows and down columns and all entries in column j are less than or equal to j. An ...

A compact manifold admits a Lorentzian structure iff its Euler characteristic vanishes. Therefore, every noncompact manifold admits a Lorentzian structure.

A group in which the elements are square matrices, the group multiplication law is matrix multiplication, and the group inverse is simply the matrix inverse. Every matrix ...

The maximum degree, sometimes simply called the maximum degree, of a graph G is the largest vertex degree of G, denoted Delta.