Spheres tangent to all the faces of a simplex

AbstractThere are at most 2n spheres tangent to all n + 1 faces of an n-simplex. It has been shown that the minimum number of such spheres is 2n − c(n, 12(n + 1)) if n is odd and 2n − c(n, 12(n + 1)) if n is even. The object of this note is to show that this result is a consequence of a theorem in graph theory

A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = \Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal and Aronov. We use this incidence bound to prove that a set of n points, no more than n-k of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy's on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.Comment: 12 page

Lines, Circles, Planes and Spheres

Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k \binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})$. For similar conditions and sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in \cite{Ell67}) we also show that the number of spheres determined by $n$ points is at least $1+\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best possible under its hypothesis. (By $t_3^{orchard}(n)$, we are denoting the maximum number of three-point lines attainable by a configuration of $n$ points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles.Comment: 37 page

Nematic-nematic phase separation in binary mixtures of thick and thin hard rods: Results from Onsager-like theories

Published versio

Neuroactive steroids in depression and anxiety disorders: Clinical studies

Certain neuroactive steroids modulate ligand-gated ion channels via non-genomic mechanisms. Especially 3 alpha-reduced pregnane steroids are potent positive allosteric modulators of the gamma-aminobutyric acid type A (GABA(A)) receptor. During major depression, there is a disequilibrium of 3 alpha-reduced neuroactive steroids, which is corrected by clinically effective pharmacological treatment. To investigate whether these alterations are a general principle of successful antidepressant treatment, we studied the impact of nonpharmacological treatment options on neuroactive steroid concentrations during major depression. Neither partial sleep deprivation, transcranial magnetic stimulation, nor electroconvulsive therapy affected neuroactive steroid levels irrespectively of the response to these treatments. These studies suggest that the changes in neuroactive steroid concentrations observed after antidepressant pharmacotherapy more likely reflect distinct pharmacological properties of antidepressants rather than the clinical response. In patients with panic disorder, changes in neuroactive steroid composition have been observed opposite to those seen in depression. However, during experimentally induced panic induction either with cholecystokinine-tetrapeptide or sodium lactate, there was a pronounced decline in the concentrations of 3 alpha-reduced neuroactive steroids in patients with panic disorder, which might result in a decreased GABAergic tone. In contrast, no changes in neuroactive steroid concentrations could be observed in healthy controls with the exception of 3 alpha,5 alpha-tetrahydrodeoxycorticosterone. The modulation of GABA(A) receptors by neuroactive steroids might contribute to the pathophysiology of depression and anxiety disorders and might offer new targets for the development of novel anxiolytic compounds. Copyright (c) 2006 S. Karger AG, Basel

Digitise This! A Quick and Easy Remote Sensing Method to Monitor the Daily Extent of Dredge Plumes

Technological advancements in remote sensing and GIS have improved natural resource managers’ abilities to monitor large-scale disturbances. In a time where many processes are heading towards automation, this study has regressed to simple techniques to bridge a gap found in the advancement of technology. The near-daily monitoring of dredge plume extent is common practice using Moderate Resolution Imaging Spectroradiometer (MODIS) imagery and associated algorithms to predict the total suspended solids (TSS) concentration in the surface waters originating from floods and dredge plumes. Unfortunately, these methods cannot determine the difference between dredge plume and benthic features in shallow, clear water. This case study at Barrow Island, Western Australia, uses hand digitising to demonstrate the ability of human interpretation to determine this difference with a level of confidence and compares the method to contemporary TSS methods. Hand digitising was quick, cheap and required very little training of staff to complete. Results of ANOSIM R statistics show remote sensing derived TSS provided similar spatial results if they were thresholded to at least 3 mg L-1. However, remote sensing derived TSS consistently provided false-positive readings of shallow benthic features as Plume with a threshold up to TSS of 6 mg L-1, and began providing false-negatives (excluding actual plume) at a threshold as low as 4 mg L-1. Semi-automated processes that estimate plume concentration and distinguish between plumes and shallow benthic features without the arbitrary nature of human interpretation would be preferred as a plume monitoring method. However, at this stage, the hand digitising method is very useful and is more accurate at determining plume boundaries over shallow benthic features and is accessible to all levels of management with basic training

From Interactions to Institutions: Microprocesses of Framing and Mechanisms for the Structuring of Institutional Fields

Despite the centrality of meaning to institutionalization, little attention has been paid to how meanings evolve and amplify to become institutionalized cultural conventions. We develop an interactional framing perspective to explain the microprocesses and mechanisms by which this occurs. We identify three amplification processes and three ways frames stack up or laminate that become the building blocks for diffusion and institutionalization of meanings within organizations and fields. Although we focus on “bottom-up” dynamics, we argue that framing occurs in a politicized social context and is inherently bidirectional, in line with structuration, because microlevel interactions instantiate macrostructures. We consider how our approach complements other theories of meaning making, its utility for informing related theoretical streams, and its implications for organizing at the meso and macro levels

Cognitive frames in corporate sustainability: managerial sensemaking with paradoxical and business case frames

Corporate sustainability confronts managers with tensions between complex economic, environmental, and social issues. Drawing on the literature on managerial cognition, corporate sustainability, and strategic paradoxes, we develop a cognitive framing perspective on corporate sustainability. We propose two cognitive frames—a business case frame and a paradoxical frame—and explore how differences between them in cognitive content and structure influence the three stages of the sensemaking process—that is, managerial scanning, interpreting, and responding with regard to sustainability issues. We explain how the two frames lead to differences in the breadth and depth of scanning, differences in issue interpretations in terms of sense of control and issue valence, and different types of responses that managers consider with regard to sustainability issues. By considering alternative cognitive frames, our argument contributes to a better understanding of managerial decision making regarding ambiguous sustainability issues, and it develops the underlying cognitive determinants of the stance that managers adopt on sustainability issues. This argument offers a cognitive explanation for why managers rarely push for radical change when faced with complex and ambiguous issues, such as sustainability, that are characterized by conflicting yet interrelated aspects