6,520 research outputs found
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 can be completed if and only if
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 , , , the result is especially simple, as
we show that any partial -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 56. The secondary revolves around
the primary in 4.9143 0.007 days at a distance of 1227 24 km,
yielding a total system mass of (6.057 0.36) 10 kg.
Combined with an IRAS size measurement, our data yield a bulk density for this
B-type asteroid of 1570 500 kg~m.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 -MnGa into (Al,Ga)As heterostructures
Electrical spin injection from ferromagnetic -MnGa into an (Al,Ga)As
p-i-n light emitting diode (LED) is demonstrated. The -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
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