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If, in the above commutative diagram of modules and module homomorphisms the columns and two upper rows are exact, then so is the bottom row.

The surface of revolution given by the parametric equations x(u,v) = cosusin(2v) (1) y(u,v) = sinusin(2v) (2) z(u,v) = sinv (3) for u in [0,2pi) and v in [-pi/2,pi/2]. It is ...

A roll of eight on a pair of six-sided dice in the game of craps. The probability of rolling "eighter from Decatur" is 5/36, or 13.888...%.

sum_(i=1)^n((partialu)/(partialx_i))^2=1.

For any Abelian group G and any natural number n, there is a unique space (up to homotopy type) such that all homotopy groups except for the nth are trivial (including the ...

A family of functors H_n(·) from the category of pairs of topological spaces and continuous maps, to the category of Abelian groups and group homomorphisms satisfies the ...

Ein(z) = int_0^z((1-e^(-t))dt)/t (1) = E_1(z)+lnz+gamma, (2) where gamma is the Euler-Mascheroni constant and E_1 is the En-function with n=1.

The Einstein field equations are the 16 coupled hyperbolic-elliptic nonlinear partial differential equations that describe the gravitational effects produced by a given mass ...

The functions E_1(x) = (x^2e^x)/((e^x-1)^2) (1) E_2(x) = x/(e^x-1) (2) E_3(x) = ln(1-e^(-x)) (3) E_4(x) = x/(e^x-1)-ln(1-e^(-x)). (4) E_1(x) has an inflection point at (5) ...

Einstein summation is a notational convention for simplifying expressions including summations of vectors, matrices, and general tensors. There are essentially three rules of ...