A treatment of atomic mean square displacement in higher order perturbation theory

A general derivation of the anharmonic coefficients for a periodic lattice invoking the special case of the central force interaction is presented. All of the contributions to mean square displacement (MSD) to order 14 perturbation theory are enumerated. A direct correspondance is found between the high temperature limit MSD and high temperature limit free energy contributions up to and including 0(14). This correspondance follows from the detailed derivation of some of the contributions to MSD. Numerical results are obtained for all the MSD contributions to 0(14) using the Lennard-Jones potential for the lattice constants and temperatures for which the Monte Carlo results were calculated by Heiser, Shukla and Cowley. The Peierls approximation is also employed in order to simplify the numerical evaluation of the MSD contributions. The numerical results indicate the convergence of the perturbation expansion up to 75% of the melting temperature of the solid (TM) for the exact calculation; however, a better agreement with the Monte Carlo results is not obtained when the total of all 14 contributions is added to the 12 perturbation theory results. Using Peierls approximation the expansion converges up to 45% of TM• The MSD contributions arising in the Green's function method of Shukla and Hubschle are derived and enumerated up to and including 0(18). The total MSD from these selected contributions is in excellent agreement with their results at all temperatures. Theoretical values of the recoilless fraction for krypton are calculated from the MSD contributions for both the Lennard-Jones and Aziz potentials. The agreement with experimental values is quite good.

Organizational commitment and perceived relatedness as correlates of the intention to continue officiating in track and field

The objectives of the present study were to explore three components of organizational commitment (affective [AC], normative [NC] and continuance [CC] commitment; Allen & Meyer, 1991), perceived relatedness (Oeci & Ryan, 1985; 2002), and behavioural intention (Ajzen, 2002) within the context of volunteer track and field officiating. The objectives were examined in a 2-phase study. Ouring phase 1, experts (N = 10) with domain familiarity assessed the item content relevance and representation of modified organizational commitment (OC; Meyer, Allen & Smith, 1993) and perceived relatedness (La Guardia, Oeci, Ryan & Couchman, 2000) items. Fourteen of 26 (p < .05) items were relevant (Aiken's coefficient V) and NC (M = 3.88, SO = .64), CC (M = 3.63, SD = .52), and relatedness (M = 4.00, SD = .93) items had mean item content-representation ratings of either "good" or "very good" while AC (M = 2.50, SD = 0.58) was rated "fair". Participants in phase 2 (N = 80) responded to items measuring demographic variables, perceptions of OC to Athletics Canada, perceived relatedness to other track and field officials, and a measure of intention (yiu, Au & Tang, 2001) to continue officiating. Internal consistency reliability estimates (Cronbach's (1951) coefficient alpha) were as follows: (a) AC = .78, (b) CC = .85, (c) NC = .80 (d) perceived relatedness = .70 and, (e) intention = .92 in the present sample. Results suggest that the track and field officials felt only minimally committed to Athletics Canada (AC M = 3.90, SD = 1.23; NC M = 2.47, SD = 1.25; CC M = 3.32; SD = 1.34) and that their relationships with other track and field officials were strongly endorsed (M = 5.86, SD = 0.74). Bivariate correlations (Pearson r) indicated that perceived relatedness to other track and field officials demonstrated the strongest relationship with intention to continue officiating (r = .346, p < .05), while dimensions of OC were not significantly related to intention (all p's > .05). Together perceived relatedness (j3 = .339, p = .004), affective commitment (j3 = -.1 53, p = .308), normative commitment (j3 = -.024, p = .864) and continuance commitment (j3 = .186, P = .287) contribute to the prediction of intention to continued officiating (K = .139). These relationships remained unaffected by the inclusion of demographic (j3age = -.02; P years with Athletics Canada = -.13; bothp's > .05) or alternative commitment (j3sport = -.19; P role = .15; Pathletes = .20; all p' s > .05) considerations. Three open-ended questions elicited qualitative responses regarding participants' reasons for officiating. Responses reflecting initial reasons for officiating formed these higher order themes: convenience, helping reasons, extension of role, and intrinsic reasons. Responses reflecting reasons for continuing to officiate formed these higher order themes: track and field, to help, and personal benefits. Responses reflecting changes that would influence continued involvement were: political, organizational/structural, and personal. These results corroborate the findings of previous investigations which state that the reasons underpinning volunteer motivations change over time (Cuskelly et al., 2002). Overall, the results of this study suggest that track and field officials feel minimal commitment to the organization of Athletics Canada but a stronger bond with their fellow officials. Moreover, the degree to which track and field officials feel meaningfully connected to one another appears to exert a positive influence on their intentions to continue officiating. As such, it is suggested that in order to promote continued involvement, Athletics Canada increases its focus on fostering environments promoting positive interactions among officials.

Molecular dynamics calculation of mean square displacement in alkali metals and rare gas solids and comparison with lattice dynamics

Molec ul ar dynamics calculations of the mean sq ua re displacement have been carried out for the alkali metals Na, K and Cs and for an fcc nearest neighbour Lennard-Jones model applicable to rare gas solids. The computations for the alkalis were done for several temperatures for temperature vol ume a swell as for the the ze r 0 pressure ze ro zero pressure volume corresponding to each temperature. In the fcc case, results were obtained for a wide range of both the temperature and density. Lattice dynamics calculations of the harmonic and the lowe s t order anharmonic (cubic and quartic) contributions to the mean square displacement were performed for the same potential models as in the molecular dynamics calculations. The Brillouin zone sums arising in the harmonic and the quartic terms were computed for very large numbers of points in q-space, and were extrapolated to obtain results ful converged with respect to the number of points in the Brillouin zone.An excellent agreement between the lattice dynamics results was observed molecular dynamics and in the case of all the alkali metals, e~ept for the zero pressure case of CSt where the difference is about 15 % near the melting temperature. It was concluded that for the alkalis, the lowest order perturbation theory works well even at temperat ures close to the melting temperat ure. For the fcc nearest neighbour model it was found that the number of particles (256) used for the molecular dynamics calculations, produces a result which is somewhere between 10 and 20 % smaller than the value converged with respect to the number of particles. However, the general temperature dependence of the mean square displacement is the same in molecular dynamics and lattice dynamics for all temperatures at the highest densities examined, while at higher volumes and high temperatures the results diverge. This indicates the importance of the higher order (eg. ~* ) perturbation theory contributions in these cases.

L-Fuzzy Relations in Coq

Heyting categories, a variant of Dedekind categories, and Arrow categories provide a convenient framework for expressing and reasoning about fuzzy relations and programs based on those methods. In this thesis we present an implementation of Heyting and arrow categories suitable for reasoning and program execution using Coq, an interactive theorem prover based on Higher-Order Logic (HOL) with dependent types. This implementation can be used to specify and develop correct software based on L-fuzzy relations such as fuzzy controllers. We give an overview of lattices, L-fuzzy relations, category theory and dependent type theory before describing our implementation. In addition, we provide examples of program executions based on our framework.