Random walks on graphs: ideas, techniques and results

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects.Comment: LateX file, 34 pages, 13 jpeg figures, Topical Revie

Symmetries of Abelian Orbifolds

Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of multiplicative sequences, one for each cycle in the Cycle Index of the permutation group. A multiplicative sequence is controlled by its values on prime numbers and their pure powers. Therefore, we pay particular attention to orbifolds of the form C^D/Gamma where the order of Gamma is p^alpha. We propose a generalization of these sequences for any D and any p.Comment: 75 pages, 13 figures, 30 table

Spherical Universe topology and the Casimir effect

The mode problem on the factored 3--sphere is applied to field theory calculations for massless fields of spin 0, 1/2 and 1. The degeneracies on the factors, including lens spaces, are neatly derived in a geometric fashion. Vacuum energies are expressed in terms of the polyhedral degrees and equivalent expressions given using the cyclic decomposition of the covering group. Scalar functional determinants are calculated and the spectral asymmetry function treated by the same approach with explicit forms on one-sided lens spaces.Comment: 33 pages, 1 figure. Typos corrected and one reference adde

On a Calculus-based Statistics Course for Life Science Students

The choice of pedagogy in statistics should take advantage of the quantitative capabilities and scientific background of the students. In this article, we propose a model for a statistics course that assumes student competency in calculus and a broadening knowledge in biology. We illustrate our methods and practices through examples from the curriculum

Microbially mediated reduction of FeIII and AsV in Cambodian sediments amended with 13C-labelled hexadecane and kerogen

Microbial activity is generally accepted to play a critical role, with the aid of suitable organic carbon substrates, in the mobilisation of arsenic from sediments into shallow reducing groundwaters. The nature of the organic matter in natural aquifers driving the reduction of AsV to AsIII is of particular importance but is poorly understood. In this study, sediments from an arsenic rich aquifer in Cambodia were amended with two 13C-labelled organic substrates. 13C-hexadecane was used as a model for potentially bioavailable long chain n-alkanes and a 13C-kerogen analogue as a proxy for non-extractable organic matter. During anaerobic incubation for 8 weeks, significant FeIII reduction and AsIII mobilisation were observed in the biotic microcosms only, suggesting that these processes were microbially driven. Microcosms amended with 13C-hexadecane exhibited a similar extent of FeIII reduction to the non-amended microcosms, but marginally higher AsIII release. Moreover, gas chromatography–mass spectrometry analysis showed that 65 % of the added 13C-hexadecane was degraded during the 8-week incubation. The degradation of 13C-hexadecane was microbially driven, as confirmed by DNA stable isotope probing (DNA-SIP). Amendment with 13C-kerogen did not enhance FeIII reduction or AsIII mobilisation, and microbial degradation of kerogen could not be confirmed conclusively by DNA-SIP fractionation or 13C incorporation in the phospholipid fatty acids. These data are, therefore, consistent with the utilisation of long chain n-alkanes (but not kerogen) as electron donors for anaerobic processes, potentially including FeIII and AsV reduction in the subsurface

Assessment of hypertension association with arsenic exposure from food and drinking water in Bihar, India

Epidemiological studies have associated chronic exposure to arsenic (As) from drinking water with increased risk of hypertension. However, evidence of an association between As exposure from food and hypertension risks is sparse. To quantify the association between daily As intake from both food (rice, wheat and potatoes) and drinking water (Aswater) along with total exposure (Astotal) and hypertension risks in a study population in Bihar, India, we conducted an individual level cross-sectional analysis between 2017 and 2019 involving 150 participants. Arsenic intake variables and three indicators of hypertension risks (general hypertension, low-density lipoprotein (LDL) and high-density lipoprotein (HDL)) were derived, and any relationship was quantified using a series of crude and multivariable log-linear or logistic regression models. The prevalence of general hypertension was 40% for the studied population. The median level of HDL was 45 mg/dL while median value of LDL was 114 mg/dL. Apart from a marginally significant positive relationship between As intake from rice and the changes of LDL (p-value = 0.032), no significant positive association between As intake and hypertension risks could be ascertained. In fact, Astotal was found to be associated with lower risks of general hypertension and higher levels of HDL (p-value = 0.020 and 0.010 respectively) whilst general hypertension was marginally associated with lower Aswater (p-value = 0.043). Due to limitations regarding study design and residual confounding, all observed marginal associations should be treated with caution

Mathematical practices and mathematical modes of enquiry: Same or different?

Background: In this paper, I share a case study of a teacher’s work on mathematics tasks in the context of a ‘mathematics for teaching’ course aiming to develop mathematical content understandings and mathematical practices among primary teachers in one South African province. The course was developed in a national context of concerns about the nature and levels of primary teachers’ mathematical knowledge. Theories viewing mathematical practices as fundamental, contrasted with those that view mathematical practices and mathematical content in more separate and ‘to be integrated’ ways, are used to frame the analysis and ritically reflect on the findings. Results: Data from this teacher’s pre-test and selected course assessments and interactions suggest that while he was able to develop some aspects of the mathematical practices described in the literature, his overall orientation remained attuned to memorization and recall. Findings also pointed to an ongoing reliance on external validation of the ‘correctness’ of his answers. Conclusions: The data suggest that the presence of elements of mathematical practices cannot be viewed as equivalent to the presence of mathematical modes of enquiry. The analysis presented in this paper suggests that while elements of mathematical practices can be developed, moving towards an encompassing orientation to mathematical modes of enquiry may require more central focus on problem-solving approaches to achieve a change in orientation

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials