Boundedness of anti-canonical volumes of singular log Fano threefolds

We prove Weak Borisov--Alexeev--Borisov Conjecture in dimension three which states that the anti-canonical volume of an $\epsilon$-klt log Fano pair of dimension three is bounded from above.Comment: 20 pages. Comments are welcome; v2: slightly modified; v3: 19 pages, proof slightly modifie

Simultaneous amplification and non-symmetric amplitude damping of two-mode Gaussian state

The evolution of two-mode Gaussian state under symmetric amplification, non-symmetric damping and thermal noise is studied. The time dependent solution of the state characteristic function is obtained. The separability criterions are given for the final state of weak amplification as well as strong amplification.Comment: 6 pages, 2 figure