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For a local non-toric Calabi–Yau manifold which arises as a smoothing of a toric Gorenstein singularity, this paper derives the open Gromov–Witten invariants of a generic fiber of the special Lagrangian fibration constructed by Gross and thereby constructs its Strominger-YauZaslow (SYZ) mirror. Moreover, it proves that the SYZ mirrors and disk potentials vary smoothly under conifold transitions, giving a global picture of SYZ mirror symmetry

It is known that every complex square matrix with nonnega-tive determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of kpositive semi-definite matrices but not fewer, for k=1, 2, 3, 4, 5.

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An asymptotic formula is given for the number of integers$n$≤$x$for which φ($n$) is not divisible by a given odd prime.