A keystone in the classical theory of diophantine approximation is the construction of an auxilliary polynomial. The polynomial is constructed so that it is forced (for arithmetic reasons) to vanish at certain approximating points and this contradicts an upper bound on the order of vanishing obtained by other (usually geometric) techniques; the contradiction then allows one to prove finiteness ...

An interesting feature of our argument is that although it uses arithmetic in an essential way, it is largely geometric; in particular there is no use of Borel’s theorem [1]. This makes Theorem 1.1 strictly stronger than [1] in this setting, since no congruence assumptions are made. We recall that a group is said to be large if it has a subgroup of finite index which maps onto a free group of r...

Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulas for the value of the continuously sampled arithmetic Asian...

In a recent pilot study, errors in forecasts of future returns showed little differences when arithmetic and geometric mean calculations served as the basis for the projections. Using annual returns on the DJIA and S&P for 1954 to 2007, 1-year-, 5-year-ahead, and 10-year-ahead forecasts were made based on 5-, 10-, 15-, and 20-year histories based on both arithmetic and geometric averages. This ...