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Discussion forums, employed by MOOC providers as the primary mode of interaction among instructors and students, have emerged as one of the important components of online courses. We empirically study contribution behavior in these online collaborative learning forums using data from 44 MOOCs hosted on Coursera, focusing primarily on the highest-volume(More)
Let M n k be the moduli space of based (anti-self-dual) instantons on CP 2 of charge k and rank n. There is a natural inclusion M n k ֒→ M n+1 k. We show that the direct limit space M ∞ k is homotopy equivalent to BU (k) × BU (k). Let ℓ∞ be a line in the complex projective plane and let CP 2 be the blow-up at a point away from ℓ∞. M n k can be alternatively(More)
We study the large n limit of the moduli spaces of Gn-instantons on S 4 and CP 2 where Gn is SU (n), Sp(n/2), or SO(n). We show that in the direct limit topology, the moduli space is homotopic to a classifying space. For example, the moduli space of Sp(∞) or SO(∞) instantons on CP 2 has the homotopy type of BU (k) where k is the charge of the instantons. We(More)
Let M (k; SO(n)) be the moduli space of based gauge equivalence classes of SO(n) instantons on principal SO(n) bundles over S 4 with rst Pontryagin class p 1 = 2k. In this paper, we use a monad description ((Ti], D]) of these moduli spaces to show that in the limit over n, the moduli space is homotopy equivalent to the classifying space BSp(k). Finally, we(More)
Since 1985 the Education Program for Gifted Youth (EPGY) at Stanford University has been developing a series of stand-alone multi-media computer-based distance-learning mathematics and physics courses from the elementary school through the university level. Because these courses are used in situations where students do not have access to regular classroom(More)
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