Results and estimates on pseudopowers

Let be a positive integer. We say looks like a power of 2 modulo a prime if there exists an integer such that . First, we provide a simple proof of the fact that a positive integer which looks like a power of modulo all but finitely many primes is in fact a power of . Next, we define an -pseudopower of the base to be a positive integer that is not a power of , but looks like a power of modulo all primes . Let denote the least such . We give an unconditional upper bound on , a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that is about for a certain constant . We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than are also given.

     

Measurement/data-processing method to improve the ruggedness of membrane-based sensors: Application to amperometric oxygen sensor

This paper describes alternative measurement and data-processing approaches that can reduce effects of experimental variables on results obtained with a membrane-based sensor for oxygen. In the new approaches, the membrane-based sensor is first equilibrated with the sample solution, after which a polarizing voltage is applied and current vs. time data are recorded as the response decays toward a steady-state condition. Current vs. time data are then processed by a fixed-time option and an integration option designed to determine the charge corresponding to the total amount of oxygen inside the membrane when a polarizing voltage is applied. The current measured at a fixed time and the total charge varied linearly with oxygen concentration between 0.05 and 0.26 mmol l(-1). Pooled relative standard deviations (N = 35) for the measurement/data-processing step were near 0.4% for the new pre-equilibrium options compared to a value of 0.3% for the steady-state option. Dependencies of the pre-equilibrium options on membrane thickness and stirring rate in the most sensitive regions were at least two orders of magnitude smaller than for the steady-state option.