Spectral Analysis for Matrix Hamiltonian Operators

In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schr\"odinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Though we focus on a proof of the 3d cubic problem, this work presents a new algorithm for verifying certain spectral properties needed to study soliton stability. Source code for verification of our comptuations, and for further experimentation, are available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.Comment: 57 pages, 22 figures, typos fixe

Scattering in flatland: Efficient representations via wave atoms

This paper presents a numerical compression strategy for the boundary integral equation of acoustic scattering in two dimensions. These equations have oscillatory kernels that we represent in a basis of wave atoms, and compress by thresholding the small coefficients to zero. This phenomenon was perhaps first observed in 1993 by Bradie, Coifman, and Grossman, in the context of local Fourier bases \cite{BCG}. Their results have since then been extended in various ways. The purpose of this paper is to bridge a theoretical gap and prove that a well-chosen fixed expansion, the nonstandard wave atom form, provides a compression of the acoustic single and double layer potentials with wave number $k$ as $O(k)$-by-$O(k)$ matrices with $O(k^{1+1/\infty})$ nonnegligible entries, with a constant that depends on the relative $\ell_2$ accuracy \eps in an acceptable way. The argument assumes smooth, separated, and not necessarily convex scatterers in two dimensions. The essential features of wave atoms that enable to write this result as a theorem is a sharp time-frequency localization that wavelet packets do not obey, and a parabolic scaling wavelength $\sim$ (essential diameter)${}^2$. Numerical experiments support the estimate and show that this wave atom representation may be of interest for applications where the same scattering problem needs to be solved for many boundary conditions, for example, the computation of radar cross sections.Comment: 39 page

Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005). The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation is firstly identified. A scale discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background (CMB) data, such as the imprint of topological defects, in particular cosmic strings, and for their reconstruction after separation from the other signal components.Comment: 22 pages, 2 figures. Version 2 matches version accepted for publication in MNRAS. Version 3 (identical to version 2) posted for code release announcement - "Steerable scale discretised wavelets on the sphere" - S2DW code available for download at http://www.mrao.cam.ac.uk/~jdm57/software.htm

A Centre-Stable Manifold for the Focussing Cubic NLS in R1+3R^{1+3}

Consider the focussing cubic nonlinear Schr\"odinger equation in $R^3$: $$i\psi_t+\Delta\psi = -|\psi|^2 \psi.$$ It admits special solutions of the form $e^{it\alpha}\phi$, where $\phi$ is a Schwartz function and a positive ($\phi>0$) solution of $$-\Delta \phi + \alpha\phi = \phi^3.$$ The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form $e^{i(v \cdot + \Gamma)} \phi(\cdot - y, \alpha)$. We prove that any solution starting sufficiently close to a standing wave in the $\Sigma = W^{1, 2}(R^3) \cap |x|^{-1}L^2(R^3)$ norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that \mc N is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones. The proof is based on the modulation method introduced by Soffer and Weinstein for the $L^2$-subcritical case and adapted by Schlag to the $L^2$-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in $R^3$ for the nonselfadjoint Schr\"odinger operator obtained by linearizing around a standing wave solution.Comment: 56 page

Data compression on the sphere

Large data-sets defined on the sphere arise in many fields. In particular, recent and forthcoming observations of the anisotropies of the cosmic microwave background (CMB) made on the celestial sphere contain approximately three and fifty mega-pixels respectively. The compression of such data is therefore becoming increasingly important. We develop algorithms to compress data defined on the sphere. A Haar wavelet transform on the sphere is used as an energy compression stage to reduce the entropy of the data, followed by Huffman and run-length encoding stages. Lossless and lossy compression algorithms are developed. We evaluate compression performance on simulated CMB data, Earth topography data and environmental illumination maps used in computer graphics. The CMB data can be compressed to approximately 40% of its original size for essentially no loss to the cosmological information content of the data, and to approximately 20% if a small cosmological information loss is tolerated. For the topographic and illumination data compression ratios of approximately 40:1 can be achieved when a small degradation in quality is allowed. We make our SZIP program that implements these compression algorithms available publicly.Comment: 13 pages, 8 figures, accepted for publication by A&A; We make our SZIP program that implements these compression algorithms available publicly from http://www.szip.org.u

Disentangling the respective contribution of task selection and task execution in self-directed cognitive control development

peer reviewedTask selection and task execution are key constructs in cognitive control development. Yet, little is known about how separable they are and how each contributes to task switching performance. Here, 60 4- to 5-year olds, 60 7- to 8-year olds, and 60 10- to 11-year olds children completed the double registration procedure, which dissociates these two processes. Task selection yielded both mixing and switch costs, especially in younger children, and task execution mostly yielded switch costs at all ages, suggesting that task selection is costlier than task execution. Moreover, both task selection and execution varied with task self-directedness (i.e., to what extent the task is driven by external aids) demands. Whereas task selection and task execution are dissociated regarding performance costs, they nevertheless both contribute to self-directed control

Examining the effect of Libet clock stimulus parameters on temporal binding

Temporal binding refers to the subjective temporal compression between actions and their outcomes. It is widely used as an implicit measure of sense of agency, that is, the experience of controlling our actions and their consequences. One of the most common measures of temporal binding is the paradigm developed by Haggard, Clark and Kalogeras (2002) based on the Libet clock stimulus. Although widely used, it is not clear how sensitive the temporal binding effect is to the parameters of the clock stimulus. Here, we present five experiments examining the effects of clock speed, number of clock markings and length of the clock hand on binding. Our results show that the magnitude of temporal binding increases with faster clock speeds, whereas clock markings and clock hand length do not significantly influence temporal binding. We discuss the implications of these results

Beyond the Libet clock: modality variants for agency measurements

The Sense of Agency (SoA) refers to our capability to control our own actions and influence the world around us. Recent research in HCI has been exploring SoA to provide users an instinctive sense of “I did that” as opposed to “the system did that”. However, current agency measurements are limited. The Intentional Binding (IB) paradigm provides an implicit measure of the SoA. However, it is constrained by requiring high visual attention to a “Libet clock” onscreen. In this paper, we extend the timing stimulus through auditory and tactile cues. Our results demonstrate that audio timing through voice commands and haptic timing through tactile cues on the hand are alternative techniques to measure the SoA using the IB paradigm. They both address limitations of the traditional method (e.g., lack of engagement and visual demand). We discuss how our results can be applied to measure SoA in tasks involving different interactive scenarios common in HCI

Archaeoseismology: Methodological issues and procedure

Archaeoseismic research contributes important data on past earthquakes. A limitation of the usefulness of archaeoseismology is due to the lack of continuous discussion about the methodology. The methodological issues are particularly important because archaeoseismological investigations of past earthquakes make use of a large variety of methods. Typical in situ investigations include: (1) reconstruction of the local archaeological stratigraphy aimed at defining the correct position and chronology of a destruction layer, presumably related to an earthquake; (2) analysis of the deformations potentially due to seismic shaking or secondary earthquake effects, detectable on walls; (3) analysis of the depositional characteristics of the collapsed material; (4) investigations of the local geology and geomorphology to define possible natural cause(s) of the destruction; (5) investigations of the local factors affecting the ground motion amplifications; and (6) estimation of the dynamic excitation, which affected the site under investigation. Subsequently, a 'territorial' approach testing evidence of synchronous destruction in a certain region may delineate the extent of the area struck by the earthquake. The most reliable results of an archaeoseismological investigation are obtained by application of modern geoarchaeological practice (archaeological stratigraphy plus geological–geomorphological data), with the addition of a geophysical-engineering quantitative approach and (if available) historical information. This gives a basic dataset necessary to perform quantitative analyses which, in turn, corroborate the archaeoseismic hypothesis. Since archaeoseismological investigations can reveal the possible natural causes of destruction at a site, they contribute to the wider field of environmental archaeology, that seeks to define the history of the relationship between humans and the environment. Finally, through the improvement of the knowledge on the past seismicity, these studies can contribute to the regional estimation of seismic hazard