Quantum Walks and Reversible Cellular Automata

We investigate a connection between a property of the distribution and a conserved quantity for the reversible cellular automaton derived from a discrete-time quantum walk in one dimension. As a corollary, we give a detailed information of the quantum walk.Comment: 15 pages, minor corrections, some references adde

Lefschetz fibrations with small slope

We construct Lefschetz fibrations over $S^2$ which do not satisfy the slope inequality. This gives a negative answer to a question of Hain.Comment: 9 pages, 2 figures, Main changes are the corrections of the statement and proof of Theorem 1.5. This version correct the statements and proofs of Corollary 1.3 and 1.