Relations, residuals, regular interiors, and relative regular equivalence

Given a relation α (a binary sociogram) and an a priori equivalence relation π, both on the same set of individuals, it is interesting to look for the largest equivalence πo that is contained in and is regular with respect to α. The equivalence relation πo is called the regular interior of π with respect to α. The computation of πo involves the left and right residuals, a concept that generalized group inverses to the algebra of relations. A polynomial-time procedure is presented (Theorem 11) and illustrated with examples. In particular, the regular interior gives meet in the lattice of regular equivalences: the regular meet of regular equivalences is the regular interior of their intersection. Finally, the concept of relative regular equivalence is defined and compared with regular equivalence.

Applications of Feller-Reuter-Riley transition functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.

What students can tell about lecturers when they are teaching: manifestations of confident and under-confident lecturers

Delivering a lecture requires confidence, a sound knowledge and well developed teaching skills (Cooper and Simonds, 2007, Quinn and Hughes, 2007). However, practitioners who are new to lecturing large groups in higher education may initially lack the confidence to do so which can manifest itself in their verbal and non-verbal cues and the fluency of their teaching skills. This results in the perception that students can identify the confident and non-confident teacher during a lecture (Street, 2007) and so potentially contributing to a lecturer’s level of anxiety prior to, and during, a lecture. Therefore, in the current educational climate of consumerisation, with the increased evaluation of teaching by students, having the ability to deliver high-quality, informed, and interesting lectures assumes greater significance for both lecturers and universities (Carr, 2007; Higher Education Founding Council 2008, Glass et al., 2006). This paper will present both the quantitative and qualitative data from a two-phase mixed method study with 75 nurse lecturers and 62 nursing students in one university in the United Kingdom. The study investigated the notion that lecturing has similarities to acting (Street, 2007). The findings presented here are concerned with how students perceived lecturers’ level of confidence and how lecturers believed they demonstrated confidence. In phase one a specifically designed questionnaire was distributed to both lecturers and students and a response rate of 91% (n=125) was achieved, while in phase two 12 in-depth semi-structured interviews were conducted with lecturers. Results suggested that students in a lecture could identify if the lecturer was confident or not by the way they performed a lecture. Students identified 57 manifestations of non-confidence and lecturers identified 85, while 57 manifestations of confidence were identified by students and 88 by lecturers. Overall, these fell into 12 main converse categories, ranging from body language to the use of space within the room. Both students and lecturers ranked body language, vocal qualities, delivery skills, involving the students and the ability to share knowledge as the most evident manifestations of confidence. Elements like good eye contact, smiling, speaking clearly and being fluent in the use of media recourses where all seen as manifestations confidence, conversely if these were poorly executed then a presentation of under confidence was evident. Furthermore, if the lecturer appeared enthusiastic it was clearly underpinned by the manifestation of a highly confidence lecturer who was secure in their knowledge base and teaching abilities: Some lecturers do appear enthusiastic but others don’t. I think the ones that do know what they are talking about, you can see it in their voice and in their lively body language. I think they are also good at involving the students even. I think the good ones are able to turn boring subjects into lively and interesting ones. (Student 50) Significantly more lecturers than students felt the lecturer should appear confident when lecturing. The lecturers stated it was particularly important to do so when they did not feel confident, because they were concerned with appearing capable. It seems that these students and lecturers perceived that expressive and apparently confident lecturers can make a positive impact on student groups in terms of involvement in lectures; the data also suggested the reverse, for the under confident lecturer. Findings from phase two indicated that these lecturers assumed a persona when lecturing, particularly, but not exclusively, when they were nervous. These lecturers went through a process of assuming and maintaining this persona before and during a lecture as a way of promoting their internal perceptions of confidence but also their outward manifestation of confidence. Although assuming a convincing persona may have a degree of deception about it, providing the knowledge communicated is accurate, the deception may aid rather than hinder learning, because enhances the delivery of a lecture. Therefore, the deception of acting a little more confidently than one feels might be justified when the lecturer knows the knowledge they are communicating is correct, unlike the Dr Fox Effect where the person delivering a lecture is an actor and does not know the subject in any detail or depth and where the deception to be justified (Naftulin, et al., 1973). In conclusion, these students and lecturers perceive that confident and enthusiastic lecturers communicate their passion for the subject in an interesting and meaningful manner through the use of their voice, body, space and interactions in such a way that shows confidence in their knowledge as well as their teaching abilities. If lecturers, therefore, can take a step back to consider how they deliver lectures in apparently confident ways this may increase their ability to engage their students and not only help them being perceived as good lecturers, but also contribute to the genuine act of education.