(4.1) Let P : R → R be a polynomial which does not vanish identically. Use Fubini’s Theorem to show, by induction on the dimension n, that Z(P ) = {x ∈ R|P (x) = 0} has Lebesgue measure equal to zero. proof By induction on n. For n = 1, consider a polynomial P ∈ R[X], and let d = degP . Then P has at most d roots, and Z(P ) = {x ∈ R|P (x) = 0} is finite, thus has measure 0. Now let n ≥ 1, and a...

In the crystal structure of the racemic title isoxazolidine, C(19)H(27)NO, the relative stereochemistry between the phenyl group and the bridgehead H atom is shown to be syn. There are two mol-ecules in the asymmetric unit, one of which is the 7R*,13R* enanti-omer, and one of which is the 7S*,13S* enanti-omer. These enanti-omers adopt different orientations of the phenyl ring with respect to th...

In this paper we give a criterion whether a given minimal symplectic 4manifold with b+2 = 1 having a torsion-free canonical class is rational or ruled. As a corollary, we confirm that most of homotopy elliptic surfaces {E(1)K |K is a fibered knot in S } constructed by R. Fintushel and R. Stern are minimal symplectic 4-manifolds with b + 2 = 1 which do not admit a complex structure.

In this project, we implemented the recursive autoencoder (RAE) as described in Socher’s paper to discover the sentiment of sentences. We train and test our RAEs with a dataset of over 10000 sentences from movie reviews, and achieve 75.4% accuracy.