On every n-long input, every two-way finite automaton (fa) can reverse its input head O(n) times before halting. A fawith few reversals is an automaton where this number is only o(n). For every h, we exhibit a language that can be recognized by an h-state nondeterministic fa with few reversals, but requires Ω(2) states on every deterministic fa with few reversals.

optimization is an important methodology for activities in planning and design. the transportation designers are able to introduce better projects when they can save time and cost of travel for project by optimization methods. most of the optimization problems in engineering are more complicated than they can be solved by custom optimization methods. the most common and available methods are he...

Unusual wave propagation effects can occur in systems with broken time-reversal symmetry. In a two-dimensional electron gas, for example, time-reversal symmetry can be broken by a perpendicular static magnetic field. As a result, the system can exhibit a quantum Hall effect, with a unidirectional flow of electrons on the edge of the system.1–4 Analogous one-way modes of photons have recently be...