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

Changes in hemlock looper [Lepidoptera: Geometridae] pupal distribution through a 3-year outbreak cycle

La distribution des chrysalides de l’arpenteuse de la pruche, Lambdina fiscellaria, a été étudiée au cours d’un cycle épidémique d’une durée de trois ans près du Lac Princeton sur l’île d’Anticosti au Québec. Au total, 10 sapins ont été coupés et toutes les chrysalides ont été comptées sur le tronc et les branches (partie non-foliée vs foliée) de la cime inférieure, médiane et supérieure, ainsi que sur le tronc sous la cime. En condition préépidémique, les chrysalides ont principalement été trouvées sur les branches des cimes médianes et supérieures. Durant l’épidémie, la densité des chrysalides n’a pas augmenté dans ces sites de pupaison et les larves se sont surtout transformées en chrysalides sur le tronc, à partir du sol jusque dans la cime médiane, ainsi que sur les branches de la cime inférieure. Peu de chrysalides ont été trouvées sur la partie foliée des branches en période post-épidémique, la plupart étant trouvées sur la partie basale non-foliée qui apparaît comme un endroit préférentiel pour la pupaison de l'arpenteuse de la pruche. De façon à optimiser la détection des augmentations de populations dans les réseaux de surveillance, des pièges à chrysalides devraient être placés à hauteur de poitrine sur le tronc de sapins baumiers.The hemlock looper, Lambdina fiscellaria, pupal distribution was studied through a 3-year outbreak cycle near Lac Princeton on Anticosti Island in Quebec. Over the 3 years, 10 balsam fir trees were cut and all pupae were counted on the stem and branches (non-foliated vs foliated parts) of the lower, middle and upper crowns and on the stem below crown. In pre-outbreak conditions, pupae were mostly found on branches of the middle and upper crowns. During the outbreak, pupal density did not increase on these parts of the trees, since pupae were mostly found on the stem, from the ground to the middle crown, and on branches of the lower crown. Few pupae were found on the foliated portion of branches in post-outbreak conditions but most were found on the basal non-foliated part of branches, which appears to be a preferred location for hemlock looper pupation. In order to optimize detection of population increases in monitoring networks, we suggest using pupal traps at breast height on balsam fir trees

Two isoperimetric inequalities for the Sobolev constant

In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalises an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.Comment: 11 page

MedZIM: Mediation analysis for Zero-Inflated Mediators with applications to microbiome data

The human microbiome can contribute to the pathogenesis of many complex diseases such as cancer and Alzheimer's disease by mediating disease-leading causal pathways. However, standard mediation analysis is not adequate in the context of microbiome data due to the excessive number of zero values in the data. Zero-valued sequencing reads, commonly observed in microbiome studies, arise for technical and/or biological reasons. Mediation analysis approaches for analyzing zero-inflated mediators are still lacking largely because of challenges raised by the zero-inflated data structure: (a) disentangling the mediation effect induced by the point mass at zero; and (b) identifying the observed zero-valued data points that are actually not zero (i.e., false zeros). We develop a novel mediation analysis method under the potential-outcomes framework to fill this gap. We show that the mediation effect of the microbiome can be decomposed into two components that are inherent to the two-part nature of zero-inflated distributions. The first component corresponds to the mediation effect attributable to a unit-change over the positive relative abundance and the second component corresponds to the mediation effect attributable to discrete binary change of the mediator from zero to a non-zero state. With probabilistic models to account for observing zeros, we also address the challenge with false zeros. A comprehensive simulation study and the applications in two real microbiome studies demonstrate that our approach outperforms existing mediation analysis approaches.Comment: Corresponding: Zhigang L

The Link between the Baryonic Mass Distribution and the Rotation Curve Shape

The observed rotation curves of disc galaxies, ranging from late-type dwarf galaxies to early-type spirals, can be fit remarkably well simply by scaling up the contributions of the stellar and HI discs. This `baryonic scaling model' can explain the full breadth of observed rotation curves with only two free parameters. For a small fraction of galaxies, in particular early-type spiral galaxies, HI scaling appears to fail in the outer parts, possibly due to observational effects or ionization of the HI. The overall success of the baryonic scaling model suggests that the well-known global coupling between the baryonic mass of a galaxy and its rotation velocity (known as the baryonic Tully-Fisher relation), applies at a more local level as well, and it seems to imply a link between the baryonic mass distribution and the distribution of total mass (including dark matter).Comment: 10 pages, accepted for publication in MNRA

Antlia Dwarf Galaxy: Distance, quantitative morphology and recent formation history via statistical field correction

We apply a statistical field correction technique originally designed to determine membership of high redshift galaxy clusters to Hubble Space Telescope imaging of the Antlia Dwarf Galaxy; a galaxy at the very edge of the Local Group. Using the tip of the red giant branch standard candle method coupled with a simple Sobel edge detection filter we find a new distance to Antlia of 1.31 +/- 0.03 Mpc. For the first time for a Local Group Member, we compute the concentration, asymmetry and clumpiness (CAS) quantitative morphology parameters for Antlia from the distribution of resolved stars in the HST/ACS field, corrected with a new method for contaminants and complement these parameters with the Gini coefficient (G) and the second order moment of the brightest 20 per cent of the flux (M_20). We show that it is a classic dwarf elliptical (C = 2.0, A = 0.063, S = 0.077, G = 0.39 and M_20 = -1.17 in the F814W band), but has an appreciable blue stellar population at its core, confirming on-going star-formation. The values of asymmetry and clumpiness, as well as Gini and M_20 are consistent with an undisturbed galaxy. Although our analysis suggests that Antlia may not be tidally influenced by NGC 3109 it does not necessarily preclude such interaction.Comment: Accepted for publication in MNRA

The ethics of digital well-being: a multidisciplinary perspective

This chapter serves as an introduction to the edited collection of the same name, which includes chapters that explore digital well-being from a range of disciplinary perspectives, including philosophy, psychology, economics, health care, and education. The purpose of this introductory chapter is to provide a short primer on the different disciplinary approaches to the study of well-being. To supplement this primer, we also invited key experts from several disciplines—philosophy, psychology, public policy, and health care—to share their thoughts on what they believe are the most important open questions and ethical issues for the multi-disciplinary study of digital well-being. We also introduce and discuss several themes that we believe will be fundamental to the ongoing study of digital well-being: digital gratitude, automated interventions, and sustainable co-well-being

Mass-to-light ratio gradients in early-type galaxy haloes

Since the near future should see a rapidly expanding set of probes of the halo masses of individual early-type galaxies, we introduce a convenient parameter for characterising the halo masses from both observational and theoretical results: \dML, the logarithmic radial gradient of the mass-to-light ratio. Using halo density profiles from LCDM simulations, we derive predictions for this gradient for various galaxy luminosities and star formation efficiencies $\epsilon_{SF}$. As a pilot study, we assemble the available \dML\ data from kinematics in early-type galaxies - representing the first unbiassed study of halo masses in a wide range of early-type galaxy luminosities - and find a correlation between luminosity and \dML, such that the brightest galaxies appear the most dark-matter dominated. We find that the gradients in most of the brightest galaxies may fit in well with the LCDM predictions, but that there is also a population of fainter galaxies whose gradients are so low as to imply an unreasonably high star formation efficiency $\epsilon_{SF} > 1$. This difficulty is eased if dark haloes are not assumed to have the standard LCDM profiles, but lower central concentrations.Comment: 17 pages, 13 figures. Accepted for publication on MNRA

GBDTMO: as new option for early-stage breast cancer detection and classification using machine learning

Breast cancer is the second leading cause of disease death in women, after lung and bronchus cancer. According to measurements, mammography misses breast cancer in 10% to 15% of cases for women aged 50 to 69 years. In the current study, we used the Wisconsin breast cancer dataset to develop a two-stage model for breast cancer diagnosis. The main goal of this study effort is to effectively carry out feature selection and classification tasks. Gradient Boosting Decision Tree-based Mayfly Optimisation (GBDTMO), an innovative and efficient breast cancer diagnostic machine learning system, is provided. In the second stage, we employ a Mayfly search to determine which subset of traits is the best. Two more well-known datasets on breast cancer, the ICCR and the Cancer Corpus, were also compared for classification accuracy. The accuracy of the suggested GBDTMO model was higher than that of the existing GBDT and Practical Federated Gradient Boosting Decision Tree (PFGBDT), which had accuracy values of 93.25% and 94.25%, respectively. Similarly, the recall, F-measure, and ROC area values were 98.52%, 97.52%, and 96.32%, respectively. Furthermore, it demonstrated a lower RMSE of 0.98 than the existing GBDT and PFGBDT