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The most general form of this theorem states that in a commutative unit ring R, the height of every proper ideal I generated by n elements is at most n. Equality is attained ...

Given a series of positive terms u_i and a sequence of finite positive constants a_i, let rho=lim_(n->infty)(a_n(u_n)/(u_(n+1))-a_(n+1)). 1. If rho>0, the series converges. ...

Let F be the set of complex analytic functions f defined on an open region containing the closure of the unit disk D={z:|z|<1} satisfying f(0)=0 and df/dz(0)=1. For each f in ...

The spherical harmonics form a complete orthogonal system, so an arbitrary real function f(theta,phi) can be expanded in terms of complex spherical harmonics by ...

A path composed of connected horizontal and vertical line segments, each passing between adjacent lattice points. A lattice path is therefore a sequence of points P_0, P_1, ...

A nonnegative measurable function f is called Lebesgue integrable if its Lebesgue integral intfdmu is finite. An arbitrary measurable function is integrable if f^+ and f^- ...

A Lehmer number is a number generated by a generalization of a Lucas sequence. Let alpha and beta be complex numbers with alpha+beta = sqrt(R) (1) alphabeta = Q, (2) where Q ...

The case of the Weierstrass elliptic function with invariants g_2=1 and g_3=0. In this case, the half-periods are given by (omega_1,omega_2)=(omega,iomega), where omega is ...

Levy (1963) noted that 13 = 3+(2×5) (1) 19 = 5+(2×7), (2) and from this observation, conjectured that all odd numbers >=7 are the sum of a prime plus twice a prime. This ...

The limit test, also sometimes known as the nth term test, says that if lima_n!=0 or this limit does not exist as n tends to infinity, then the series suma_n does not ...