Torsional rigidity for cylinders with a Brownian fracture

We obtain bounds for the expected loss of torsional rigidity of a cylinder $\Omega_L=(-L/2,L/2) \times \Omega\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $\Omega_L,$ and runs until the first time it exits $\Omega_L$. These bounds are expressed in terms of the geometry of the cross-section $\Omega \subset \R^2$. It is shown that if $\Omega$ is a disc with radius $R$, then in the limit as $L \rightarrow \infty$ the expected loss of torsional rigidity equals $cR^5$ for some $c\in (0,\infty)$. We derive bounds for $c$ in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in $\R^3$ with radius $1,$ and runs until the first time it exits this ball.Comment: 18 page

Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.Comment: 18 pages, 4 figure

Heat content and inradius for regions with a Brownian boundary

In this paper we consider $\beta[0; s]$, Brownian motion of time length $s > 0$, in $m$-dimensional Euclidean space $\mathbb R^m$ and on the $m$-dimensional torus $\mathbb T^m$. We compute the expectation of (i) the heat content at time $t$ of $\mathbb R^m\setminus \beta[0; s]$ for fixed $s$ and $m = 2,3$ in the limit $t \downarrow 0$, when $\beta[0; s]$ is kept at temperature 1 for all $t > 0$ and $\mathbb R^m\setminus \beta[0; s]$ has initial temperature 0, and (ii) the inradius of $\mathbb R^m\setminus \beta[0; s]$ for $m = 2,3,\cdots$ in the limit $s \rightarrow \infty$.Comment: 13 page

PEARL model for pesticide behaviour and emissions in soil-plant systems : description of the processes in FOCUS PEARL v 1.1.1

The use of pesticides in agriculture presents risks to the environment, which are increasingly evaluated by using computation models. The new PEARL model simulates the behaviour of pesticides in soil-plant systems and their emissions to the environment. The pesticide model is used in combination with the hydrological model SWAP. Various agricultural situations and ways of applying the pesticides can be simulated. The model accounts for different sorption mechanisms, in equilibrium and non-equilibriumdomains of the soil. Pesticide transport in the liquid and gas phases is described by the convection-dispersion-diffusion type equation, which is supplemented with sink terms. Comprehensive reaction schemes are processed in matrix form. The rate in first-order transformation kinetics is dependent on temperature, soil moisture content and depth in the soil. Besides computing persistence and distribution of the pesticidal compounds in soil, the model computes volatilization into the air, lateral drainage to water courses and leaching to groundwater

Factors contributing to the negation of therapeutic services by emerging adults at a South African university

From a practice point of view it seems as if there are certain factors that might contribute to the fact that emerging adults tend to negate therapeutic help and services. It also seems to be specifically true with regard to emerging adults at university. Help negation seems to occur albeit the fact that therapeutic intervention is seen as an effective tool in managing distress. The aim of the study therefore was to explore which factors contribute to help negation behaviour in emerging adults at a specific university in South Africa.A qualitative case study design was employed where participants who complied with the inclusion criteria set out for the study, were selected by means of non-probability target and snowball sampling. Fifteen students residing in campus residences respectively participated in one of three focus group discussions. Creswell’s spiral of data analysis was used to analyse the transcribed data.The data crystallised into four themes, which contribute to help negation behaviour in emerging adults. Emerging adults have a fear of being judged, stigmatised, recognised, of not being treated confidentially and a fear of being vulnerable and hurt. They prefer to seek help and support from their family and peers and to put their trust in God and their religion; they have internalised beliefs about themselves and therapy and a need for independence; student interns work at the therapeutic centres on campus which specifically contributes to help negation for emerging adults at the university. It is crucial that the professionals revisit their strategies and approaches in order to overcome negation of formal help and create a more understandable, approachable and effective therapeutic service to emerging adults, especially at university therapeutic centres

’n Verkennende studie met die oog op die identifísering en kategorisering van leerteorieë met verwysing na die prediking

An introductory study to identify and classify theories of learning with regard to the task of preachingLearning is a lifelong process in which man must be what he can be, namely a being interacting with his world in a creative problem-solving manner for the well-being of himself and others. In a similar sense the church has always seen her task in preaching, supported by all the other domains of churchlife, as that of teaching people to come to terms with the gospel of Jesus Christ in their daily existence. This article proposes to identify, categorize and integrate the acknowledged theories underlyingthe learning process, as they exist in the social sciences, into an all encompassingmodel for learning; a model from which conclusions are drawn in the hope that further studies can spell out the implications of these conclusions as they are applicable to the task of preaching within the church

On the minimization of Dirichlet eigenvalues of the Laplace operator

We study the variational problem \inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \}, where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, \h(\partial \Omega) is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega|$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $\Omega_{2,k}$. If $m \ge 2$, and if $\Omega_{m,k}$ is a minimiser, then $\Omega_{m,k}^*:= \textup{int}(\overline{\Omega_{m,k}})$ is also a minimiser, and $\R^m\setminus \Omega_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $\Omega_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $\Omega_{m,k}$ has at most $4$ components. Furthermore $\Omega_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.Comment: 16 page

Photometric variability in the old open cluster M 67. II. General Survey

We use differential CCD photometry to search for variability in BVI among 990 stars projected in and around the old open cluster M 67. In a previous paper we reported results for 22 cluster members that are optical counterparts to X-ray sources; this study focuses on the other stars in our observations. A variety of sampling rates were employed, allowing variability on time scales ranging from \sim 0.3 hours to \sim 20 days to be studied. Among the brightest sources studied, detection of variability as small as sigma approx 10 mmag is achieved (with > 3 sigma confidence); for the typical star observed, sensitivity to variability at levels sigma approx 20 mmag is achieved. The study is unbiased for stars with 12.5 < B < 18.5, 12.5 < V < 18.5, and 12 < I < 18 within a radius of about 10 arcmin from the cluster centre. In addition, stars with 10 < BVI < 12.5 were monitored in a few small regions in the cluster. We present photometry for all 990 sources studied, and report the variability characteristics of those stars found to be variable at a statistically significant level. Among the variables, we highlight several sources that merit future study, including stars located on the cluster binary sequence, stars on the giant branch, blue stragglers, and a newly discovered W UMa system.Comment: 12 pages, including 6 figures and 5 tables. Tables 1 and 3 only available in electronic version of paper. Accepted by A&

Effects of plant pathogens on population dynamics and community composition in grassland ecosystems: two case studies

Grassland ecosystems comprise a major portion of the earth’s terrestrial surface, ranging from high-input cultivated monocultures or simple species mixtures to relatively unmanaged but dynamic systems. Plant pathogens are a component of these systems with their impact dependent on many interacting factors, including grassland species population dynamics and community composition, the topics covered in this paper. Plant pathogens are affected by these interactions and also act reciprocally by modifying their nature. We review these features of disease in grasslands and then introduce the 150-year long-term Park Grass Experiment (PGE) at Rothamsted Research in the UK. We then consider in detail two plant-pathogen systems present in the PGE, Tragopogon pratensis-Puccinia hysterium and Holcus lanata-Puccinia coronata. These two systems have very different life history characteristics: the first, a biennial member of the Asteraceae infected by its host-specific, systemic rust; the second, a perennial grass infected by a host-non-specific rust. We illustrate how observational, experimental and modelling studies can contribute to a better understanding of population dynamics, competitive interactions and evolutionary outcomes. With Tragopogon pratensis-Puccinia hysterium, characterised as an “outbreak” species in the PGE, we show that pathogen-induced mortality is unlikely to be involved in host population regulation; and that the presence of even a short-lived seed-bank can affect the qualitative outcomes of the host-pathogen dynamics. With Holcus lanata-Puccinia coronata, we show how nutrient conditions can affect adaptation in terms of host defence mechanisms, and that co-existence of competing species affected by a common generalist pathogen is unlikely