Weierstrass’ theorem with weights

25 pages, no figures.-- MSC2000 codes: 41A10, 41A28, 41A30.MR#: MR2053535 (2005g:41082)Zbl#: Zbl 1045.41005We characterize the set of functions which can be approximated by continuous functions in the L∞ norm with respect to almost every weight. This allows to characterize the set of functions which can be approximated by polynomials or by smooth functions for a wide range of weights.Research by first, third and fourth authors partially supported by a grant from DGI (BFM 2000-0022), Spain. Research by third author also partially supported by a grant from DGI (BFM 2000-0206-C04-01), Spain.Publicad

Flopping and slicing: SO(4) and Spin(4)-models

We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graphs A₁. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell–Weil group of the elliptic fibration is required to be trivial for Spin(4) and ℤ/2ℤ for SO(4). Each of these models has two possible crepant resolutions connected by a flop. We also compute a generating function for the Euler characteristic of such elliptic fibrations over a base of arbitrary dimensions. In the case of a threefold, we also compute the triple intersection numbers of the fibral divisors. In the case of Calabi–Yau threefolds, we also compute their Hodge numbers and check the cancellations of anomalies in a six-dimensional supergravity theory