On the structure of P(n)*P(n) for p=2

We show that P(n)*(P(n)) for p = 2 with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation Epsilon nor the coproduct Delta are multiplicative. As a consequence the algebra structure of P(n)*(P(n)) is slightly different from what was supposed to be the case. We give formulas for Epsilon(xy) and Delta(xy) and show that the inversion of the formal group of P(n) is induced by an antimultiplicative involution Xi : P(n) -> P(n). Some consequences for multiplicative and antimultiplicative automorphisms of K(n) for p = 2 are also discussed

Optical Maser Emission from Trivalent Praseodymium in Calcium Tungstate

Coherent emission at 1.047 µ from trivalent praseodymium in calcium tungstate was observed. This emission coincides with strong infrared fluorescence at the same wavelength and was found to be stimulated mostly by absorption of blue light by the 3P0, 3P1, and 3P2 bands. The emission corresponds to a 1G4-->3H4 transition with the terminal level 377 cm^–1 from the ground state. The oscillation threshold was the same at 4.2°, 20°, and 78°K. No stimulated emission was observed at room temperature. The lifetime of the metastable state 1G4 is 50×10^–3 sec. A new technique used to measure the lifetime is described

Computing a minimal resolution over the Steenrod algebra

We describe an algorithm that allows to compute a minimal resolution of the Steenrod algebra. The algorithm has built-in knowledge about vanishing lines for the cohomology of sub Hopf algebras of the Steenrod algebra which makes it both faster and more economical than the generic approach