Quartic Forms in Many Variables

We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms $132$ variables are sufficient.Comment: 8 page

Two applications of the curve lemma for orthogonal groups

We show (under some hypothesis in small dimensions) that the analytic degree of the divisor of a modular form on the orthogonal group O(2,p) is determined by its weight. Moreover, we prove that certain integrals, occurring in Arakelov intersection theory, associated with modular forms on O(2,p) converge.Comment: 16 page

pp-adic Zeros of Quintic Forms

It is shown that a quintic form over a p-adic field with at least 26 variables has a non-trivial zero, providing that the cardinality of the residue class field exceeds 9.Comment: Minor correction

The effect of the amido substituent on polymer molecular weight in propene homopolymerisation by titanium cyclopentadienyl-amide catalysts

In the homopolymerisation of propene by the cyclopentadienyl-amide titanium catalyst systems [η5,η1-C5H4(CH2)2NR]TiCl2/MAO and [η5,η1-C5H4(CH2)2NR]Ti(CH2Ph)2/B(C6F5)3 (R = tBu, iPr, Me), the catalyst with the smallest substituent (Me) on the amido moiety consistently gives the highest polymer molecular weight. This differs from the trend usually observed in related catalysts with tetramethylcyclopentadienyl-amide ancillary ligands, where larger amide substituents result in higher molecular weights. Based on the present information a hypothesis is formulated in which an increased cation-anion interaction for the less sterically hindered catalyst is responsible for disfavouring chain transfer relative to chain growth.

On Borcherds products associated with lattices of prime discriminant

We show that certain spaces of vector valued modular forms are isomorphic to spaces of scalar valued modular forms whose Fourier coefficients are supported on suitable progressions. As an application we give a very explicit description of Borcherds products on Hilbert modular surfaces.Comment: To appear in Ramanujan