Termination of (many) 4-dimensional log flips

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(K+D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and termination of flips in a relative birational setting. We also prove termination of directed flips with big K+D. As a consequence, we prove existence of minimal models of 4-dimensional dlt pairs of general type, existence of 5-dimensional log flips, and rationality of Kodaira energy in dimension 4.Comment: 13 pages; a minor change in the proof of Thm.4.

Pulse Control of Decoherence with Population Decay

The pulse control of decoherence in a qubit interacting with a quantum environment is studied with focus on a general case where decoherence is induced by both pure dephasing and population decay. To observe how the decoherence is suppressed by periodic pi pulses, we present a simple method to calculate the time evolution of a qubit under arbitrary pulse sequences consisting of bit-flips and/or phase-flips. We examine the effectiveness of the two typical sequences: bb sequence consisting of only bit-flips, and bp sequence consisting of both bit- and phase-flips. It is shown that the effectiveness of the pulse sequences depends on a relative strength of the two decoherence processes especially when a pulse interval is slightly shorter than qubit-environment correlation times. In the short-interval limit, however, the bp sequence is always more effective than, or at least as effective as, the bb sequence.Comment: 11 pages, 7 figure