Density Matrix Renormalisation Group Calculations for Two-Dimensional Lattices: An Application to the Spin-Half and Spin-One Square-Lattice Heisenberg Models

A new density matrix renormalisation group (DMRG) approach is presented for quantum systems of two spatial dimensions. In particular, it is shown that it is possible to create a multi-chain-type 2D DMRG approach which utilises previously determined system and environment blocks {\it at all points}. One firstly builds up effective quasi-1D system and environment blocks of width $L$ and these quasi-1D blocks are then used to as the initial building-blocks of a new 2D infinite-lattice algorithm. This algorithm is found to be competitive with those results of previous 2D DMRG algorithms and also of the best of other approximate methods. An illustration of this is given for the spin-half and spin-one Heisenberg models on the square lattice. The best results for the ground-state energies per bond of the spin-half and spin-one square-lattice Heisenberg antiferromagnets for the $N = 20 \times 20$ lattice using this treatment are given by $E_g/N_B = -0.3321$ and $E_g/N_B = -1.1525$, respectively.Comment: 7 Figures. Accepted for publication in Phys. Rev.

An exploration of pathologies of multilevel principal components analysis in statistical models of shape

3D facial surface imaging is a useful tool in dentistry and in terms of diagnostics and treatment planning. Between-group PCA (bgPCA) is a method that has been used to analyse shapes in biological morphometrics, although various “pathologies” of bgPCA have recently been proposed. Monte Carlo (MC) simulated datasets were created here in order to explore “pathologies” of multilevel PCA (mPCA), where mPCA with two levels is equivalent to bgPCA. The first set of MC experiments involved 300 uncorrelated normally distributed variables, whereas the second set of MC experiments used correlated multivariate MC data describing 3D facial shape. We confirmed results of numerical experiments from other researchers that indicated that bgPCA (and so also mPCA) can give a false impression of strong differences in component scores between groups when there is none in reality. These spurious differences in component scores via mPCA decreased significantly as the sample sizes per group were increased. Eigenvalues via mPCA were also found to be strongly affected by imbalances in sample sizes per group, although this problem was removed by using weighted forms of covariance matrices suggested by the maximum likelihood solution of the two-level model. However, this did not solve problems of spurious differences between groups in these simulations, which was driven by very small sample sizes in one group. As a “rule of thumb” only, all of our experiments indicate that reasonable results are obtained when sample sizes per group in all groups are at least equal to the number of variables. Interestingly, the sum of all eigenvalues over both levels via mPCA scaled approximately linearly with the inverse of the sample size per group in all experiments. Finally, between-group variation was added explicitly to the MC data generation model in two experiments considered here. Results for the sum of all eigenvalues via mPCA predicted the asymptotic amount for the total amount of variance correctly in this case, whereas standard “single-level” PCA underestimated this quantity

Dental and maxillofacial radiology: confidence, knowledge and skills in the newly graduated dentist

Introduction The UK General Dental Council says that the new graduate will have the range of skills required to begin working as part of a dental team and be well-prepared for independent practice. This study examines the views of new dental graduates in the area of dental and maxillofacial radiology. Materials and methods A questionnaire survey of newly graduated dentists in Wales and South West England asked about their experience of undergraduate education in dental and maxillofacial radiology, and their confidence in the use and interpretation of imaging techniques in early independent practice. Results Most survey participants were confident in the teaching they had received in dental radiography and radiology, and that they could apply their knowledge and understanding to the more common investigations used in general dental practice. They were less confident in the uncommonly used techniques, such as oblique lateral and lateral cephalometric views. Discussion This study is encouraging in that new dental graduates say that their undergraduate education has given them the knowledge to be able to take, interpret and make clinical diagnoses using the commonest radiological investigations. This tends to validate the undergraduate education received and identifies areas to improve. Conclusion New dental graduates are confident in their knowledge and skills, and apply these to the most commonly used radiological investigations used in general dental practice

Initial Investigations of the cranial size and shape of adult Eurasian otters (Lutra lutra) in Great Britain

Three-dimensional (3D) surface scans were carried out in order to determine the shapes of the upper sections of (skeletal) crania of adult Eurasian otters (Lutra lutra) from Great Britain. Landmark points were placed on these shapes using a graphical user interface (GUI) and distance measurements (i.e., the length, height, and width of the crania) were found by using the landmark points. Male otters had significantly larger skulls than females (P < 0.001). Differences in size also occurred by geographical area in Great Britain (P < 0.05). Multilevel Principal Components Analysis (mPCA) indicated that sex and geographical area explained 31.1% and 9.6% of shape variation in “unscaled” shape data and that they explained 17.2% and 9.7% of variation in “scaled” data. The first mode of variation at level 1 (sex) correctly reflected size changes between males and females for “unscaled” shape data. Modes at level 2 (geographical area) also showed possible changes in size and shape. Clustering by sex and geographical area was observed in standardized component scores. Such clustering in a cranial shape by geographical area might reflect genetic differences in otter populations in Great Britain, although other potentially confounding factors (e.g., population age-structure, diet, etc.) might also drive regional differences. This work provides a successful first test of the effectiveness of 3D surface scans and multivariate methods, such as mPCA, to study the cranial morphology of otters

The impact of a reduction in fluoride concentration in the Malaysian water supply on the prevalence of fluorosis and dental caries

Objective To assess the prevalence and severity of dental fluorosis and caries among Malaysian children following the reduction in fluoride concentration from 0.7 to 0.5 parts per million (ppm) in the public water supply. Methods This study involved lifelong residents aged 9‐ and 12‐year‐olds in fluoridated and nonfluoridated areas in Malaysia (n = 1155). In the fluoridated area, children aged 12 years and 9 years were exposed to 0.7 and 0.5 ppm, respectively, at the times when maxillary central incisors developed. Standardized photographs of maxillary central incisors were blind scored for fluorosis using Dean's criteria. Dental caries was examined using ICDAS‐II criteria. Results The prevalence of fluorosis (Dean's score ≥ 2) among children in the fluoridated area (35.7%, 95% CI: 31.9%‐39.6%) was significantly higher (P < 0.001) than children in the nonfluoridated area (5.5%, 95% CI: 3.6%‐7.4%). Of those in the fluoridated area, the prevalence of fluorosis decreased from 38.4% (95% CI: 33.1%‐44.3%) for 12‐year‐olds to 31.9% (95% CI: 27.6%‐38.2%) for 9‐year‐olds, although this difference was not statistically significant (P = 0.139). The mean caries experience in the permanent dentition was significantly lower in the fluoridated area than in the nonfluoridated area for both age groups (P < 0.05). In the multivariate models, the difference in the differences of caries experience between fluoridated and nonfluoridated areas remained statistically significant. This suggests that caries‐preventive effect is still maintained at 0.5 ppm. Conclusion Findings indicate that the change in fluoride level from 0.7 to 0.5 ppm has reduced fluorosis and maintains a caries‐preventive effect. Although there is a reduction in fluorosis prevalence, the difference was not statistically significant

Ultrasound-guided fine-needle aspiration cytology of lesions in the head and neck performed without local anaesthesia – An analysis of pain perception

Introduction Ultrasound-guided fine-needle aspiration cytology is commonly carried out in the head and neck. The aim was to examine the amount of pain experienced by patients undergoing this procedure carried out without the use of local anaesthetic. Methods A questionnaire was given to 109 consecutive patients undergoing ultrasound-guided fine-needle aspiration cytology containing a visual analogue scale. Patients were asked to mark with a vertical line on the 100 mm horizontal scale amount of pain they experienced during the biopsy. The pain was subsequently categorised as ‘no pain’, ‘mild pain’, ‘moderate pain’ or ‘severe pain’ based on previous pain studies. Results Hundred patients completed the visual analogue scale section of the questionnaire satisfactorily. Twenty-one patients experienced no pain, 62 experienced mild pain and 17 experienced moderate pain. No patients experienced severe pain. Further analysis showed that females had significantly higher visual analogue scale scores (Mann–Whitney test: U = 925.5, z = 2.211, P = 0.027). Patients who were aware that they were going to have a biopsy had significantly lower visual analogue scale scores than those who were not aware (Mann–Whitney test: U = 859.5, z = 2.263, P = 0.024). Conclusions Ultrasound-guided fine-needle aspiration cytology is generally a well-tolerated procedure with pain scores being higher in females. It is advised that patients are told by the referring clinician the need for biopsy as this reduces the amount of pain experienced. Keywords: Biopsy, ultrasound, pai

Evaluation of a questionnaire to measure the impact of mathematics masterclasses at the University of South Wales

The Royal Institution began the mathematics masterclasses in order to foster mathematics outreach activities from academia to the wider community. Here we validate a questionnaire that has previously been used to measure the impact of these masterclasses held at the University of South Wales (formerly known as the University of Glamorgan) in 2010 and 2011. We administered questionnaires to 111 key-stage 3 (KS3) school children and to 76 key-stage 4 (KS4) school children recruited from local schools in the south Wales area. Cronbach’s α coefficient was found to be greater than 0.7, thus indicating that the questionnaires were internally consistent. Exploratory factor analysis demonstrated that considerable structure occurred in the responses from both the KS3 and KS4 students. The questionnaire was broken down into five subsections or “subscales” (made up of related questions), namely: enthusiasm and (separately) confidence in mathematics; future plans of the students involving mathematics; course delivery and content; and, perceived usefulness of mathematics in society. Results for the confidence and usefulness subscales were found to be high for both student groups. A reduced level of enthusiasm (adequate only) was seen for the KS3 students, which might have been due to unrealistic expectations for the masterclasses prior to the course in this cohort of very young students or it might have been a characteristic of this particular cohort of students. An overall questionnaire score indicated that the masterclasses were generally well-received by the students, which makes sense because only the most mathematically able students were selected to participate by their schools, although individual items and subscales gave a more nuanced picture. All of these results indicate that the questionnaire was a viable tool that gave sensible results for our cohort of students. Potential refinements to the questionnaire are suggested