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

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

Do the Emotional Benefits of Optimism Vary Across Older Adulthood? A Life-Span Perspective

Objective. This study examined whether the emotional benefits of dispositional optimism for managing stressful encounters decrease across older adulthood. Such an effect might emerge because age-related declines in opportunities for overcoming stressors could reduce the effectiveness of optimism. Method. This hypothesis was tested in a six-year longitudinal study of 171 community-dwelling older adults (age range = 64 to 90 years). Results. Hierarchical linear models showed that dispositional optimism protected relatively young participants from exhibiting elevations in depressive symptoms over time, but that these benefits became increasingly reduced among their older counterparts. Moreover, the findings showed that an age-related association between optimism and depressive symptoms was observed particularly during periods of enhanced, as compared to reduced, stress. Conclusions. These results suggest that dispositional optimism protects emotional well-being during the early phases of older adulthood, but that its effects are reduced in advanced old age

Glafenine-induced intestinal injury in zebrafish is ameliorated by -opioid signaling via enhancement of Atf6-dependent cellular stress responses

Beside their analgesic properties, opiates exert beneficial effects on the intestinal wound healing response. In this study, we investigated the role of μ-opioid receptor (MOR) signaling on the unfolded protein response (UPR) using a novel zebrafish model of NSAID-induced intestinal injury. The NSAID glafenine was administered to zebrafish larvae at 5 days post-fertilization (dpf) for up to 24 hours in the presence or absence of the MOR-specific agonist DALDA. By analysis with histology, transmission electron microscopy and vital dye staining, glafenine-treated zebrafish showed evidence of endoplasmic reticulum and mitochondrial stress, with disrupted intestinal architecture and halted cell stress responses, alongside accumulation of apoptotic intestinal epithelial cells in the lumen. Although the early UPR marker BiP was induced with glafenine-induced injury, downstream atf6 and s-xbp1 expression were paradoxically not increased, explaining the halted cell stress responses. The μ-opioid agonist DALDA protected against glafenine-induced injury through induction of atf6-dependent UPR. Our findings show that DALDA prevents glafenine-induced epithelial damage through induction of effective UPR

The Bosma effect revisited - I. HI and stellar disc scaling models

The observed proportionality between the centripetal contribution of the dynamically insignificant HI gas in the discs of spiral galaxies and the dominant contribution of DM - the "Bosma effect" - has been repeatedly mentioned in the literature but largely ignored. We have re-examined the evidence for the Bosma effect by fitting Bosma effect models for 17 galaxies in the THINGS data set, either by scaling the contribution of the HI gas alone or by using both the observed stellar disc and HI gas as proxies. The results are compared with two models for exotic cold DM: internally consistent cosmological NFW models with constrained compactness parameters, and URC models using fully unconstrained Burkert density profiles. The Bosma models that use the stellar discs as additional proxies are statistically nearly as good as the URC models and clearly better than the NFW ones. We thus confirm the correlation between the centripetal effects of DM and that of the interstellar medium of spiral galaxies. The edificacy of "maximal disc" models is explained as the natural consequence of "classic" Bosma models which include the stellar disc as a proxy in regions of reduced atomic gas. The standard explanation - that the effect reflects a statistical correlation between the visible and exotic DM - seems highly unlikely, given that the geometric forms and hence centripetal signatures of spherical halo and disc components are so different. A literal interpretation of the Bosma effect as being due to the presence of significant amounts of disc DM requires a median visible baryon to disc DM ratio of about 40%.Comment: Accepted by A&A (Paper I

Solid state image sensor research, phase 2

Solid state image sensor arra

Requirements for Trustworthy Artificial Intelligence – A Review

The field of algorithmic decision-making, particularly Artificial Intelligence (AI), has been drastically changing. With the availability of a massive amount of data and an increase in the processing power, AI systems have been used in a vast number of high-stake applications. So, it becomes vital to make these systems reliable and trustworthy. Different approaches have been proposed to make theses systems trustworthy. In this paper, we have reviewed these approaches and summarized them based on the principles proposed by the European Union for trustworthy AI. This review provides an overview of different principles that are important to make AI trustworthy

Commensal microbiota stimulate systemic neutrophil migration through induction of Serum amyloid A: Microbiota regulate systemic neutrophil function

Neutrophils serve critical roles in inflammatory responses to infection and injury, and mechanisms governing their activity represent attractive targets for controlling inflammation. The commensal microbiota is known to regulate the activity of neutrophils and other leucocytes in the intestine, but the systemic impact of the microbiota on neutrophils remains unknown. Here we utilized in vivo imaging in gnotobiotic zebrafish to reveal diverse effects of microbiota colonization on systemic neutrophil development and function. The presence of a microbiota resulted in increased neutrophil number and myeloperoxidase expression, and altered neutrophil localization and migratory behaviours. These effects of the microbiota on neutrophil homeostasis were accompanied by an increased recruitment of neutrophils to injury. Genetic analysis identified the microbiota-induced acute phase protein serum amyloid A (Saa) as a host factor mediating microbial stimulation of tissue-specific neutrophil migratory behaviours. In vitro studies revealed that zebrafish cells respond to Saa exposure by activating NF-κB, and that Saa-dependent neutrophil migration requires NF-κB-dependent gene expression. These results implicate the commensal microbiota as an important environmental factor regulating diverse aspects of systemic neutrophil development and function, and reveal a critical role for a Saa-NF-κB signalling axis in mediating neutrophil migratory responses

Microbial Colonization Induces Dynamic Temporal and Spatial Patterns of NF-κB Activation in the Zebrafish Digestive Tract

The nuclear factor κ-light-chain enhancer of activated B cells (NF-κB) transcription factor pathway is activated in response to diverse microbial stimuli to regulate expression of genes involved in immune responses and tissue homeostasis. However, the temporal and spatial activation of NF-κB in response to microbial signals have not been determined in whole living organisms, and the molecular and cellular details of these responses are not well understood. We used in vivo imaging and molecular approaches to analyze NF-κB activation in response to the commensal microbiota in transparent gnotobiotic zebrafish