Zero-sum problems in finite cyclic groups

The purpose of this thesis is to investigate some open problems in the area of combinatorial number theory referred to as zero-sum theory. A zero-sequence in a finite cyclic group G is said to have the basic property if it is equivalent under group automorphism to one which has sum precisely IGI when this sum is viewed as an integer. This thesis investigates two major problems, the first of which is referred to as the basic pair problem. This problem seeks to determine conditions for which every zero-sequence of a given length in a finite abelian group has the basic property. We resolve an open problem regarding basic pairs in cyclic groups by demonstrating that every sequence of length four in Zp has the basic property, and we conjecture on the complete solution of this problem. The second problem is a 1988 conjecture of Kleitman and Lemke, part of which claims that every sequence of length n in Zn has a subsequence with the basic property. If one considers the special case where n is an odd integer we believe this conjecture to hold true. We verify this is the case for all prime integers less than 40, and all odd integers less than 26. In addition, we resolve the Kleitman-Lemke conjecture for general n in the negative. That is, we demonstrate a sequence in any finite abelian group isomorphic to Z2p (for p ~ 11 a prime) containing no subsequence with the basic property. These results, as well as the results found along the way, contribute to many other problems in zero-sum theory.

The effects of simple procedural, cognitive procedural and declarative learning on sleep /

Sleep spindles have been found to increase following an intense period of learning on a combination of motor tasks. It is not clear whether these changes are task specific, or a result of learning in general. The current study investigated changes in sleep spindles and spectral power following learning on cognitive procedural (C-PM), simple procedural (S-PM) or declarative (DM) learning tasks. It was hypothesized that S-PM learning would result in increases in Sigma power during Non-REM sleep, whereas C-PM and DM learning would not affect Sigma power. It was also hypothesized that DM learning would increase Theta power during REM sleep, whereas S-PM and C-PM learning would not affect Theta power. Thirty-six participants spent three consecutive nights in the sleep laboratory. Baseline polysomnographic recordings were collected on night 2. Participants were randomly assigned to one of four conditions: C-PM, S-PM, DM or control (C). Memory task training occurred on night 3 followed by polysomnographic recording. Re-testing on respective memory tasks occurred one-week following training. EEG was sampled at 256Hz from 16 sites during sleep. Artifact-free EEG from each sleep stage was submitted to power spectral analysis. The C-PM group made significantly fewer errors, the DM group recalled more, and the S-PM improved on performance from test to re-test. There was a significant night by group interaction for the duration of Stage 2 sleep. Independent t-tests revealed that the S-PM group had significantly more Stage 2 sleep on the test night than the C group. The C-PM and the DM group did not differ from controls in the duration of Stage 2 sleep on test night. There was no significant change in the duration of slow wave sleep (SWS) or REM sleep. Sleep spindle density (spindles/minute) increased significantly from baseline to test night following S-PM learning, but not for C-PM, DM or C groups. This is the first study to have shown that the same pattern of results was found for spindles in SWS. Low Sigma power (12-14Hz) increased significantly during SWS following S-PM learning but not for C-PM, DM or C groups. This effect was maximal at Cz, and the largest increase in Sigma power was at Oz. It was also found that Theta power increased significantly during REM sleep following DM learning, but not for S-PM, C-PM or C groups. This effect was maximal at Cz and the largest change in Theta power was observed at Cz. These findings are consistent with the previous research that simple procedural learning is consolidated during Stage 2 sleep, and provide additional data to suggest that sleep spindles across all non-REM stages and not just Stage 2 sleep may be a mechanism for brain plasticity. This study also provides the first evidence to suggest that Theta activity during REM sleep is involved in memory consolidation.