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Quotient sets have attracted the attention of mathe- maticians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gauss- ian primes is also dense in the complex plane. Sittinger has proved that the set of quotients of primes in an imaginary quadratic ring is dense in the complex plane and the set of quotients of primes in a real quadratic number ring is dense in R: An interesting open question is introduced by Sittinger: Is the set of quotients of Hurwitz primes dense in the quaternions? In this paper, we answer the question and prove that the set of all quotients of Hurwitz primes is dense in the quaternions.

Weak mirror symmetry relates a manifold with complex structure to another manifold equipped with a symplectic structure through a quasiisomorphism of associated differential Gerstenhaber algebras. The two manifolds are then mirror partners. In this paper we consider the analogous problem on Lie algebras. In particular we show that the semi-direct product of a Lie algebra equipped with a torsion-free flat connection with itself is a mirror partner of a semi-direct product of the same Lie algebra with its dual space. For nilpotent algebras this analysis on Lie algebras can be applied to the compact quotients of the underlying nilpotent group. We classify mirror pairs among 6-dimensional nilpotent Lie algebras that have the semi-direct product structure as well as mirror pairs admitting the more involved special Lagrangian structure, namely compatible complex and symplectic structures on the same space.

In this paper we present a new algorithm to attack lattice based cryptosystems by solving a problem over real numbers. In the case of the NTRU cryptosystem, if we assume the additional information on the modular operations, we can break the NTRU cryptosystems completely by getting the secret key. We believe that this fact was not known before.