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.

Mass spectral studies of some pentafluorophenyl derivatives of group V /|nby Andrew Tanner Rake. -- 260 St. Catharines, Ont. : [s. n.],

The mass spectra of compounds of t he series (C6F5 )3-n MP~ (n = 1,2,3, M = P and As ), (C6F5>3Sb, Ph) Sb and (C6F5 )2SbPh have been studied in detail and the important modes of fragmentation were e1ucidated, a ided by metastable ions. Various trends attributed to the central atom and or the . substituent groups have been noted and, where applicable, compared to recent studies on related phenyl and pentafluorophenyl compounds of groups IV and V. The mass spectra of fluorine containing organometallic compounds exhibit characteristic migrations of fluorine to t he central atom, giving an increasing abundance of MF+, MF2+' and RMF+ (R = Ph or C6F5) ions on descending the group_ The mass spectra of pentafluorophenyl , antimony, and arsenic compounds show a greater fragmentation of the aromatic ring than those of phosphorus. The mixed phenyl pentafluorophenyl derivatives show a characteristic pattern depending on the number of phenyl grm.lps present but show t he general characteristics of both the tris(phenyl) and tris(pentafluorophenyl) compounds. The diphenyl pentafluorophenyl der ivatives show the loss of biphenyl ion as the most import ant step, the los s of phenyl t o give the i on PhMC6F5 + being of secondary importance. The ,bis(pentafluorophenyl) phenyl derivatives fragment primarily by loss of PhC6F5 to give C6F5M+ ions, the abundance of t hese increasing r apidly from phosphorus to arsenic. This species then, exhibits a characteristic fragmentation observed in the tris(penta- fluorophenyl ) compounds. However, the abundance of (C6F5)2M+ species in these compounds i s small. I ons of the type C6H4MC6F4 + and tetrafluorobiphenylene ions C6H4C6F4 + also are observed on substitution of a phenyl group for a penta- fluorophenyl group. The fully fluorinated species (C6F4)2M+ is not observed, although octafluorobiphenylene ions , (C6F4)2+' are evident in several spectra . The appearance potentials of the major ions were obtatned from the ionisation efficiency curves. Attempts were made to correlate these to the effect of the central atom in substituent groups, but the large errors involved prevented the reaching of quantitative conclusions, although it would appear that the electron is removed from the ligand in the ionisation of t he parent molecule .

The finite bandwidth model for spin fluctuations in Pd

The frequency dependence of the electron-spin fluctuation spectrum, P(Q), is calculated in the finite bandwidth model. We find that for Pd, which has a nearly full d-band, the magnitude, the range, and the peak frequency of P(Q) are greatly reduced from those in the standard spin fluctuation theory. The electron self-energy due to spin fluctuations is calculated within the finite bandwidth model. Vertex corrections are examined, and we find that Migdal's theorem is valid for spin fluctuations in the nearly full band. The conductance of a normal metal-insulator-normal metal tunnel junction is examined when spin fluctuations are present in one electrode. We find that for the nearly full band, the momentum independent self-energy due to spin fluctuations enters the expression for the tunneling conductance with approximately the same weight as the self-energy due to phonons. The effect of spin fluctuations on the tunneling conductance is slight within the finite bandwidth model for Pd. The effect of spin fluctuations on the tunneling conductance of a metal with a less full d-band than Pd may be more pronounced. However, in this case the tunneling conductance is not simply proportional to the self-energy.

Investigating Changes in Parent Knowledge about Obsessive Compulsive Behaviour Following Group Function-Based Cognitive Behavioural Therapy for Children with High Functioning Autism

This study investigated improvements in parent knowledge of effective intervention strategies following participation in a group function-based CBT treatment (GFbCBT) package for children with comorbid OCD and ASD. Nineteen parents of children ages 7-12 years with High Functioning Autism (HFA) participated in the 9-week treatment program. Key components of treatment included psychoeducation and mapping, cognitive-behavioural skills training, function-based interventions and exposure and response prevention (ERP). Treatment sessions also included direct parent education, which followed a behavioural skills training model (Miltenberger, 2008). Parent knowledge (N = 19) was measured pre and post treatment using a vignette about a child demonstrating obsessive-compulsive behaviour. Results of a one-tailed pairwise t-test indicated statistically significant changes (p=.036) in overall parent knowledge following participation in treatment. Statistically significant changes were also found in parents’ ability to generate ERP and function-based intervention strategies. These results provide preliminary evidence that parents benefit from active involvement in the GFbCBT treatment package.

Rings, Group Rings, and Their Graphs

We associate some graphs to a ring R and we investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of the graphs associated to R. Let Z(R) be the set of zero-divisors of R. We define an undirected graph ᴦ(R) with nonzero zero-divisors as vertices and distinct vertices x and y are adjacent if xy=0 or yx=0. We investigate the Isomorphism Problem for zero-divisor graphs of group rings RG. Let Sk denote the sphere with k handles, where k is a non-negative integer, that is, Sk is an oriented surface of genus k. The genus of a graph is the minimal integer n such that the graph can be embedded in Sn. The annihilating-ideal graph of R is defined as the graph AG(R) with the set of ideals with nonzero annihilators as vertex such that two distinct vertices I and J are adjacent if IJ=(0). We characterize Artinian rings whose annihilating-ideal graphs have finite genus. Finally, we extend the definition of the annihilating-ideal graph to non-commutative rings.

The effect of synchronized group activities on pain threshold as a predictor of cooperation

Recent research suggests that participating in vigorous synchronized physical activity may result in elevated levels of endorphins, which may in turn affect social bonding (Cohen et. al., 2009). The present research aimed to examine whether or not the change in pain tolerance would be able to predict participants’ willingness to cooperate after statistically controlling for the groups’ condition. Participants were asked to run on a treadmill for 30 minutes under one of two conditions (control vs. synchronized). Prior to and after the run participants underwent a pain tolerance test. Once completed, a second activity was introduced to the participants; a cooperative game. A public goods game was used to measure an individual’s willingness to cooperate. The results showed the synchronized condition was able to predict that participants cooperated more during the public goods game (p = .009), however the change in pain threshold was unable to significantly predict cooperation (p = .32).

Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudo-conformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schr¨odinger equations to group-invariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of one-dimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether’s theorem, all conservation laws arising from these point symmetry groups are listed. Some group-invariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative two-dimensional form of the Schr¨odinger equations involving an extra modulation term with a parameter m = 2−n = 0 is discussed.