Successive projections on hyperplanes

AbstractAny sequence of points in Rn obtained by successive projections of a point on elements of a finite set of hyperplanes is bounded

Convex Independence in Permutation Graphs

A set C of vertices of a graph is P_3-convex if every vertex outside C has at most one neighbor in C. The convex hull \sigma(A) of a set A is the smallest P_3-convex set that contains A. A set M is convexly independent if for every vertex x \in M, x \notin \sigma(M-x). We show that the maximal number of vertices that a convexly independent set in a permutation graph can have, can be computed in polynomial time

Predicting the effects of deep brain stimulation using a reduced coupled oscillator model

This is the final version. Available on open access from Public Library of Science via the DOI in this recordData Availability: The data analysed in this manuscript is available from MRC BNDU Data Sharing platform at: https://data.mrc.ox.ac.uk/data-set/tremor-data-measured-essential-tremor-patients-subjected-phase-locked-deep-brain DOI: 10.5287/bodleian:xq24eN2KmDeep brain stimulation (DBS) is known to be an effective treatment for a variety of neurological disorders, including Parkinson’s disease and essential tremor (ET). At present, it involves administering a train of pulses with constant frequency via electrodes implanted into the brain. New ‘closed-loop’ approaches involve delivering stimulation according to the ongoing symptoms or brain activity and have the potential to provide improvements in terms of efficiency, efficacy and reduction of side effects. The success of closed-loop DBS depends on being able to devise a stimulation strategy that minimizes oscillations in neural activity associated with symptoms of motor disorders. A useful stepping stone towards this is to construct a mathematical model, which can describe how the brain oscillations should change when stimulation is applied at a particular state of the system. Our work focuses on the use of coupled oscillators to represent neurons in areas generating pathological oscillations. Using a reduced form of the Kuramoto model, we analyse how a patient should respond to stimulation when neural oscillations have a given phase and amplitude, provided a number of conditions are satisfied. For such patients, we predict that the best stimulation strategy should be phase specific but also that stimulation should have a greater effect if applied when the amplitude of brain oscillations is lower. We compare this surprising prediction with data obtained from ET patients. In light of our predictions, we also propose a new hybrid strategy which effectively combines two of the closed-loop strategies found in the literature, namely phase-locked and adaptive DBS

The Partial Visibility Representation Extension Problem

For a graph $G$, a function $\psi$ is called a \emph{bar visibility representation} of $G$ when for each vertex $v \in V(G)$, $\psi(v)$ is a horizontal line segment (\emph{bar}) and $uv \in E(G)$ iff there is an unobstructed, vertical, $\varepsilon$-wide line of sight between $\psi(u)$ and $\psi(v)$. Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph $G$, a bar visibility representation $\psi$ of $G$, additionally, puts the bar $\psi(u)$ strictly below the bar $\psi(v)$ for each directed edge $(u,v)$ of $G$. We study a generalization of the recognition problem where a function $\psi'$ defined on a subset $V'$ of $V(G)$ is given and the question is whether there is a bar visibility representation $\psi$ of $G$ with $\psi(v) = \psi'(v)$ for every $v \in V'$. We show that for undirected graphs this problem together with closely related problems are \NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Sub-wavelength surface IR imaging of soft-condensed matter

Outlined here is a technique for sub-wavelength infrared surface imaging performed using a phase matched optical parametric oscillator laser and an atomic force microscope as the detection mechanism. The technique uses a novel surface excitation illumination approach to perform simultaneously chemical mapping and AFM topography imaging with an image resolution of 200 nm. This method was demonstrated by imaging polystyrene micro-structures

Visibility Representations of Boxes in 2.5 Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $\Gamma$ is the set of bottom faces of the boxes in $\Gamma$.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Preparation and characterization of polypyrrole/graphene nanocomposite films and their electrochemical performance

A one-step electrochemical process had been employed to synthesize nanocomposite films of polypyrrole/graphene (PPy/GR) by electrochemical polymerisation on indium tin oxide (ITO) from an aqueous solution containing pyrrole monomer, graphene oxide (GO) nanosheets and sodium p-toluenesulfonate (NapTS). The X-ray diffraction (XRD) patterns showed that the typical peak of GO at 9.9o was missing from the nanocomposite’s diffraction pattern, suggesting that the GO had been stripped off of its oxygenous groups after the reaction. We postulated that a nanocomposite film was produced through a layer-by-layer deposition based on field emission scanning electron microscope (FESEM) images. The Raman spectroscopy profiles exhibited that the D/G intensity ratio (ID/IG) of PPy was not altered by the inclusion of GO due to the low concentration of the material used. However, the concentration was sufficient to increase the specific capacitance of the nanocomposite by 20 times compared to that of pure PPy, reflecting a synergistic effect between PPy and GR, as analysed by a three-electrode electrochemical cell. The electrochemical performance of the nanocomposites was affected by varying the deposition parameters such as concentrations of pyrrole and GO, scan rate, deposition time and deposition potential