The Hyperplane is the Only Stable, Smooth Solution to the Isoperimetric Problem in Gaussian Space

We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $\Sigma^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We show that such $\Sigma$ satisfy a curvature condition, and derive the Jacobi operator $L$ for the second variation of such $\Sigma$. Our first main result is that for non-planar $\Sigma$, bounds on the index of $L$, acting on volume preserving variations, gives us that $\Sigma$ splits off a linear space. A corollary of this result is that hyperplanes are the only stable smooth complete solutions to this Gaussian isoperimetric type problem, and that there are no hypersurfaces of index one. Finally, we show that for the case of $\Sigma^2 \subset \mathbb R^3$, there is a gradient decay estimate depending on bounds for the curvature condition and an appropriate area growth bound. This shows that, in the limit as $R \to \infty$, stable $(\Sigma, \partial\Sigma) \subset (B_{2R}(0), \partial B_{2R}(0))$ with good area growth bounds approach hyperplanes.Comment: 20 page

Invisible Market: Energy and Agricultural Technologies for Women's Economic Advancement

This research explores what it takes for technology initiatives, specifically in the energy and agricultural sectors, to reach and economically benefit women in developing countries through market-based strategies that have the potential for achieving scale and financial sustainability. It builds on ICRW's landmark paper, Bridging the Gender Divide: How Technology Can Advance Women Economically, which made the case for how technologies can create pathways for strengthening women's economic opportunities. Through a field-level investigation and interviews with experts, the authors examine how women's use of technology and their involvement in the development and distribution of a technology can not only advance women economically, but also can benefit enterprise-based technology initiatives by expanding their markets and helping them generate greater financial returns

Admiral Nurse Standards: an overview and application of ‘values based practice’

This short presentation will offer an introduction to Values Based Practice through the prism of the Admiral Nurse Standards. It is intended to demonstrate that an understanding of ‘values’ is critical to the delivery of excellent nursing care. In addition, the presentation intends to illustrate that working with values is not a theoretical and ‘airy’ idea, but rather a key practical skill that can be mastered by nurses at various levels. The presentation seeks to engage participants in a debate on whether working positively with values is a key feature of high level leadership and practice and is worthy of detailed examination

A Transnational History of a Writer in Four Packages

This is the final version. Available on open access from Taylor & Francis via the DOI in this recordArts and Humanities Research Council (AHRC