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A group G is nilpotent if the upper central sequence 1=Z_0<=Z_1<=Z_2<=...<=Z_n<=... of the group terminates with Z_n=G for some n. Nilpotent groups have the property that ...

If two curves phi and psi of multiplicities r_i!=0 and s_i!=0 have only ordinary points or ordinary singular points and cusps in common, then every curve which has at least ...

The north pole is the point on a sphere with maximum z-coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is ...

The inverse curve for a parabola given by x = at^2 (1) y = 2at (2) with inversion center (x_0,y_0) and inversion radius k is x = x_0+(k(at^2-x_0))/((at^2+x_0)^2+(2at-y_0)^2) ...

A curve given by the Cartesian equation b^2y^2=x^3(a-x). (1) It has area A=(a^3pi)/(8b). (2) The curvature is kappa(x)=(2b^2(3a^2-12ax+8x^2))/(sqrt(x)[4b^2(a-x)+(3a-4x)^2x]). ...

A positive measure is a measure which is a function from the measurable sets of a measure space to the nonnegative real numbers. Sometimes, this is what is meant by measure, ...

Let pi be a unitary representation of a group G on a separable Hilbert space, and let R(pi) be the smallest weakly closed algebra of bounded linear operators containing all ...

Let alpha be a nonzero rational number alpha=+/-p_1^(alpha_1)p_2^(alpha_2)...p_L^(alpha_L), where p_1, ..., p_L are distinct primes, alpha_l in Z and alpha_l!=0. Then ...

Given a vector space V, its projectivization P(V), sometimes written P(V-0), is the set of equivalence classes x∼lambdax for any lambda!=0 in V-0. For example, complex ...

A proper fraction is a fraction p/q with p,q>0 such that p/q<1. A fraction p/q>1 is called an improper fraction.