arXiv subjects

Ian Wood

Publications and source records attributed to Ian Wood.

24 records · Page 2Linked to original sources

arXiv2015

Gap localization of TE-Modes by arbitrarily weak defects

This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. It is shown that arbitrarily weak perturbations introduce spectrum into the spectral gaps of the background operator.

math-ph↗

arXiv2014

Designing a minimalist socially aware robotic agent for the home

We present a minimalist social robot that relies on long timeseries of low resolution data such as mechanical vibration, temperature, lighting, sounds and collisions. Our goal is to develop an experimental system for growing socially situated robotic agents whose behavioral repertoire is subsumed by the social order of the space. To get there we are designing robots that use their simple sensors and motion feedback routines to recognize different classes of human activity and then associate to each class a range of appropriate behaviors. We use the Katie Family of robots, built on the iRobot Create platform, an Arduino Uno, and a Raspberry Pi. We describe its sensor abilities and exploratory tests that allow us to develop hypotheses about what objects (sensor data) correspond to something known and observable by a human subject. We use machine learning methods to classify three social scenarios from over a hundred experiments, demonstrating that it is possible to detect social situations with high accuracy, using the low-resolution sensors from our minimalist robot.

cs.RO↗

arXiv2012

On the spectrum of waveguides in planar photonic bandgap structures

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a two-dimensional periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. The perturbation is expected to introduce guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that, somewhat unexpectedly, guided mode spectrum can be created by arbitrarily "small" perturbations. Secondly we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum $k_x$ the perturbation generates at most finitely many new eigenvalues inside the gap.

math-ph↗

arXiv2011

(Non-)Equivalence of Universal Priors

Ray Solomonoff invented the notion of universal induction featuring an aptly termed "universal" prior probability function over all possible computable environments. The essential property of this prior was its ability to dominate all other such priors. Later, Levin introduced another construction --- a mixture of all possible priors or `universal mixture'. These priors are well known to be equivalent up to multiplicative constants. Here, we seek to clarify further the relationships between these three characterisations of a universal prior (Solomonoff's, universal mixtures, and universally dominant priors). We see that the the constructions of Solomonoff and Levin define an identical class of priors, while the class of universally dominant priors is strictly larger. We provide some characterisation of the discrepancy.

cs.IT↗

arXiv2009

The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction $A_B$ of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces $\Sc$ and $\tilde{\Sc}$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples -- one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line -- which together indicate that the abstract results are probably best possible.

math.SP↗

arXiv2007

Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map.

math.SP↗