Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Assessing the psychometric properties of dream content questionnaires

The present study was the first of its kind to systematically explore the psychometric properties of dream content questionnaires as measures of dream experience. One hundred and six University students filled out the Dream Content Questionnaire (DCQ) and kept a 14-day dream diary on two separate occasions, in addition to filling out the NEO-PI-R and Multidimensional Personality Questionnaire and measures of spatial ability and imaginativeness. The DCQ's reliability was acceptable, as was its discriminant and construct validity. Six of eight predicted relationships between trait personality and DCQ reported dream content were significant. In contrast, dream diaries showed instability over time and were unrelated to personality traits. The DCQ's concurrent validity could not be adequately appraised due to the inconsistency in dream diary content over time. The results suggest that questionnaires may be used to measure dream experience; however, the precise utility of dream questionnaires remains unclear. The findings raise important questions concerning measures of dream experience.

An Investigation of the Role of Attention in the Cross-Race Effect: An Ecological Approach

The current set of studies was conducted to examine the cross-race effect (CRE), a phenomenon commonly found in the face perception literature. The CRE is evident when participants display better own-race face recognition accuracy than other-race recognition accuracy (e.g. Ackerman et al., 2006). Typically the cross-race effect is attributed to perceptual expertise, (i.e., other-race faces are processed less holistically; Michel, Rossion, Han, Chung & Caldara, 2006), and the social cognitive model (i.e., other-race faces are processed at the categorical level by virtue of being an out-group member; Hugenberg, Young, Bernstein, & Sacco, 2010). These effects may be mediated by differential attention. I investigated whether other-race faces are disregarded and, consequently, not remembered as accurately as own-race (in-group) faces. In Experiment 1, I examined how the magnitude of the CRE differed when participants learned individual faces sequentially versus when they learned multiple faces simultaneously in arrays comprising faces and objects. I also examined how the CRE differed when participants recognized individual faces presented sequentially versus in arrays of eight faces. Participants’ recognition accuracy was better for own-race faces than other-race faces regardless of familiarization method. However, the difference between own- and other-race accuracy was larger when faces were familiarized sequentially in comparison to familiarization with arrays. Participants’ response patterns during testing differed depending on the combination of familiarization and testing method. Participants had more false alarms for other-race faces than own-race faces if they learned faces sequentially (regardless of testing strategy); if participants learned faces in arrays, they had more false alarms for other-race faces than own-races faces if ii i they were tested with sequentially presented faces. These results are consistent with the perceptual expertise model in that participants were better able to use the full two seconds in the sequential task for own-race faces, but not for other-race faces. The purpose of Experiment 2 was to examine participants’ attentional allocation in complex scenes. Participants were shown scenes comprising people in real places, but the head stimuli used in Experiment 1 were superimposed onto the bodies in each scene. Using a Tobii eyetracker, participants’ looking time for both own- and other-race faces was evaluated to determine whether participants looked longer at own-race faces and whether individual differences in looking time correlated with individual differences in recognition accuracy. The results of this experiment demonstrated that although own-race faces were preferentially attended to in comparison to other-race faces, individual differences in looking time biases towards own-race faces did not correlate with individual differences in own-race recognition advantages. These results are also consistent with perceptual expertise, as it seems that the role of attentional biases towards own-race faces is independent of the cognitive processing that occurs for own-race faces. All together, these results have implications for face perception tasks that are performed in the lab, how accurate people may be when remembering faces in the real world, and the accuracy and patterns of errors in eyewitness testimony.

Phonon spectra and temperature variation of thermodynamic properties of fcc metals via Finnis-Sinclair type many body potentials: Sutton-Chen and improved Sutton-Chen models

Volume(density)-independent pair-potentials cannot describe metallic cohesion adequately as the presence of the free electron gas renders the total energy strongly dependent on the electron density. The embedded atom method (EAM) addresses this issue by replacing part of the total energy with an explicitly density-dependent term called the embedding function. Finnis and Sinclair proposed a model where the embedding function is taken to be proportional to the square root of the electron density. Models of this type are known as Finnis-Sinclair many body potentials. In this work we study a particular parametrization of the Finnis-Sinclair type potential, called the "Sutton-Chen" model, and a later version, called the "Quantum Sutton-Chen" model, to study the phonon spectra and the temperature variation thermodynamic properties of fcc metals. Both models give poor results for thermal expansion, which can be traced to rapid softening of transverse phonon frequencies with increasing lattice parameter. We identify the power law decay of the electron density with distance assumed by the model as the main cause of this behaviour and show that an exponentially decaying form of charge density improves the results significantly. Results for Sutton-Chen and our improved version of Sutton-Chen models are compared for four fcc metals: Cu, Ag, Au and Pt. The calculated properties are the phonon spectra, thermal expansion coefficient, isobaric heat capacity, adiabatic and isothermal bulk moduli, atomic root-mean-square displacement and Gr\"{u}neisen parameter. For the sake of comparison we have also considered two other models where the distance-dependence of the charge density is an exponential multiplied by polynomials. None of these models exhibits the instability against thermal expansion (premature melting) as shown by the Sutton-Chen model. We also present results obtained via pure pair potential models, in order to identify advantages and disadvantages of methods used to obtain the parameters of these potentials.