An analogue of Ryser's Theorem for partial Sudoku squares

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can be completed if and only if $R$ satisfies Hall's Condition for partial latin squares. We define Hall's Condition for partial Sudoku squares and show that Hall's Condition for partial Sudoku squares gives a criterion for the completion of partial Sudoku rectangles that is both necessary and sufficient. In the particular case where $n=pq$, $p|r$, $q|s$, the result is especially simple, as we show that any $r \times s$ partial $(p,q)$-Sudoku rectangle can be completed (no further condition being necessary).Comment: 19 pages, 10 figure

Mass and density of B-type asteroid (702) Alauda

Observations with the adaptive optics system on the Very Large Telescope reveal that outer main belt asteroid (702) Alauda has a small satellite with primary to secondary diameter ratio of $\sim$56. The secondary revolves around the primary in 4.9143 $\pm$ 0.007 days at a distance of 1227 $\pm$ 24 km, yielding a total system mass of (6.057 $\pm$ 0.36) $\times$ 10$^{18}$ kg. Combined with an IRAS size measurement, our data yield a bulk density for this B-type asteroid of 1570 $\pm$ 500 kg~m$^{-3}$.Comment: In press, ApJ 2011. 6 pages, 4 figure

On the basic character of residue classes

Let t, b be mutually prime positive integers. We say that the residue class t mod b is basic if theie exists n such that tn ≡1 mod b; otherwise t is not basic. In this paper we relate the basic character of t mod b to the quadratic character of t modulo the prime factors of b. If all prime factors p of b satisfy p ≡ 3 mod 4, then t is basic mod b if t is a quadratic nonresidue mod p for all such p ; and t is not basic mod b if t is a quadratic residue mod p for all such p. If, for all prime factors p of b, p ≡ 1 mod 4 and t is a quadratic non-residue mod p, the situation is more complicated. We define d(p) to be the highest power of 2 dividing (p - 1) and postulate that d(p) takes the same value for all prime factors p of b. Then t is basic mod b. We also give an algorithm for enumerating the (prime) numbers p lying in a given residue class mod 4t and satisfying d(p) = d. In an appendix we briefly discuss the case when b is even

Spin injection from perpendicular magnetized ferromagnetic δ\delta-MnGa into (Al,Ga)As heterostructures

Electrical spin injection from ferromagnetic $\delta$-MnGa into an (Al,Ga)As p-i-n light emitting diode (LED) is demonstrated. The $\delta$-MnGa layers show strong perpendicular magnetocrystalline anisotropy, enabling detection of spin injection at remanence without an applied magnetic field. The bias and temperature dependence of the spin injection are found to be qualitatively similar to Fe-based spin LED devices. A Hanle effect is observed and demonstrates complete depolarization of spins in the semiconductor in a transverse magnetic field.Comment: 4 pages, 3 figure

Softening of the insulating phase near Tc for the photo-induced insulator-to-metal phase transition in vanadium dioxide

We use optical-pump terahertz-probe spectroscopy to investigate the near-threshold behavior of the photoinduced insulator-to-metal (IM) transition in vanadium dioxide thin films. Upon approaching Tc a reduction in the fluence required to drive the IM transition is observed, consistent with a softening of the insulating state due to an increasing metallic volume fraction (below the percolation limit). This phase coexistence facilitates the growth of a homogeneous metallic conducting phase following superheating via photoexcitation. A simple dynamic model using Bruggeman effective medium theory describes the observed initial condition sensitivity.Comment: accepted for publication in Physical Review Letter