In its Fifth Assessment Report, the Intergovernmental Panel on Climate Change, a group of 1,300 independent scientific experts from countries all over the world under the auspices of the United Nations, concluded there's a more than 95 percent probability that human activities over the past 50 years have warmed our planet.
Scalar multiplication of two vectors (to give the so-called dot product) is commutative (i.e., aÂ·b = bÂ·a), but vector multiplication (to give the cross product) is not (i.e., a Ã b = âb Ã a). The commutative law does not necessarily hold for multiplication of conditionally convergent series.
Problem 1: Let R be a commutative unital ring, and let S be a multiplicative submonoid of R. Define an equivalence relation ~ on R x S by (a,s)~(b,t) if there is r in S with rat = rbs. Let a/s denote the ~-equivalence class of (a,s). Show that with a/s + b/t = (at+bs)/st and (a/s)(b/t) = ab/st