We give several new criteria for a quasi-projective variety to be affine. In particular, we prove that an algebraic manifold Y with dimension n is affine if and only if H(Y,ΩjY ) = 0 for all j ≥ 0, i > 0 and κ(D,X) = n, i.e., there are n algebraically independent nonconstant regular functions on Y , where X is the smooth completion of Y , D is the effective boundary divisor with support X−Y and...

In this article we consider a Brownian motion with drift, denoted by S = (St)t≥0, of the form dSt = μtdt+ dBt for t ≥ 0, with a specific non-trivial drift predictable with respect to F , the natural filtration of the Brownian motion B = (Bt)t≥0. We construct a process H = (Ht)t≥0 also predictable with respect to F B such that ((H · S)t)t≥0 is a Brownian motion in its own filtration. Furthermore...