Embedded Eigenvalues and the Nonlinear Schrodinger Equation

A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances. The proof is computer assisted as it depends on the sign of certain inner products which do not readily admit analytic representations. Our source code is available for verification at http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.Comment: 29 pages, 27 figures: fixed a typo in an equation from the previous version, and added two equations to clarif

Sensing and seeing associated with overlapping occipitoparietal activation in simultaneous EEG-fMRI.

The presence of a change in a visual scene can influence brain activity and behavior, even in the absence of full conscious report. It may be possible for us to sense that such a change has occurred, even if we cannot specify exactly where or what it was. Despite existing evidence from electroencephalogram (EEG) and eye-tracking data, it is still unclear how this partial level of awareness relates to functional magnetic resonance imaging (fMRI) blood oxygen level dependent (BOLD) activation. Using EEG, fMRI, and a change blindness paradigm, we found multi-modal evidence to suggest that sensing a change is distinguishable from being blind to it. Specifically, trials during which participants could detect the presence of a colour change but not identify the location of the change (sense trials), were compared to those where participants could both detect and localise the change (localise or see trials), as well as change blind trials. In EEG, late parietal positivity and N2 amplitudes were larger for localised changes only, when compared to change blindness. However, ERP-informed fMRI analysis found no voxels with activation that significantly co-varied with fluctuations in single-trial late positivity amplitudes. In fMRI, a range of visual (BA17,18), parietal (BA7,40), and mid-brain (anterior cingulate, BA24) areas showed increased fMRI BOLD activation when a change was sensed, compared to change blindness. These visual and parietal areas are commonly implicated as the storage sites of visual working memory, and we therefore argue that sensing may not be explained by a lack of stored representation of the visual display. Both seeing and sensing a change were associated with an overlapping occipitoparietal network of activation when compared to blind trials, suggesting that the quality of the visual representation, rather than the lack of one, may result in partial awareness during the change blindness paradigm

Exploring the biological and chemical complexity of the ligases.

Using a novel method to map and cluster chemical reactions, we have re-examined the chemistry of the ligases [Enzyme Commission (EC) Class 6] and their associated protein families in detail. The type of bond formed by the ligase can be automatically extracted from the equation of the reaction, replicating the EC subclass division. However, this subclass division hides considerable complexities, especially for the C-N forming ligases, which fall into at least three distinct types. The lower levels of the EC classification for ligases are somewhat arbitrary in their definition and add little to understanding their chemistry or evolution. By comparing the multi-domain architecture of the enzymes and using sequence similarity networks, we examined the links between overall reaction and evolution of the ligases. These show that, whilst many enzymes that perform the same overall chemistry group together, both convergent (similar function, different ancestral lineage) and divergent (different function, common ancestor) evolution of function are observed. However, a common theme is that a single conserved domain (often the nucleoside triphosphate binding domain) is combined with ancillary domains that provide the variation in substrate binding and function

Inhomogeneous magnetism in single crystalline Sr3_3CuIrO6+δ_{6+\delta}: Implications to phase-separation concepts

The single crystalline form of an insulator, Sr$_3$CuIrO$_{6+\delta}$, is shown to exhibit unexpectedly more than one magnetic transition (at 5 and 19 K) with spin-glass-like magnetic susceptibility behaviour. On the basis of this finding, viz., inhomogeneous magnetism in a chemically homogeneous material, we propose that the idea of "phase- separation" described for manganites [1] is more widespread in different ways. The observed experimental features enable us to make a comparison with the predictions of a recent toy model [2] on {\it magnetic} phase separation in an insulating environment.Comment: 4 pages, 4 figure

Pan-aortic hybrid treatment of mega-aorta syndrome

Hybrid procedures combining traditional open and newer endovascular techniques are increasingly used to treat complex aortic disease. We present a novel approach for total aortic replacement, including hybrid repair of the arch and thoracoabdominal aorta, in a patient with “mega-aorta syndrome.” A two-stage approach using a valve-sparing aortic root replacement, total arch replacement (stage I elephant trunk), and left carotid-axillary bypass was used to treat the root, proximal-mid arch, and left subclavian aneurysmal pathology. This was followed by a hybrid distal arch/Extent II thoracoabdominal aneurysm repair 3 months later. After 15 months follow-up, the patient remains asymptomatic with an intact repair, no endoleak, and normal ventricular and aortic valve function. This case demonstrates a novel “pan-aortic” hybrid approach for repair of extensive thoracic aortic disease

Religious education and hermeneutics: the case of teaching about Islam

This article attempts to do three things: the first is an exploration of the ways in which Islam is presented in an essentialist way (with a focus on Religious Education (RE) in England and Wales), leading to stereotypes and unsubstantiated generalisations. Secondly, it provides a critique of essentialism, and finally a case is made for the role of hermeneutics in the teaching and learning of Islam

Wide-field LOFAR-LBA power-spectra analyses: Impact of calibration, polarization leakage and ionosphere

Contamination due to foregrounds (Galactic and Extra-galactic), calibration errors and ionospheric effects pose major challenges in detection of the cosmic 21 cm signal in various Epoch of Reionization (EoR) experiments. We present the results of a pilot study of a field centered on 3C196 using LOFAR Low Band (56-70 MHz) observations, where we quantify various wide field and calibration effects such as gain errors, polarized foregrounds, and ionospheric effects. We observe a `pitchfork' structure in the 2D power spectrum of the polarized intensity in delay-baseline space, which leaks into the modes beyond the instrumental horizon (EoR/CD window). We show that this structure largely arises due to strong instrumental polarization leakage ($\sim30\%$) towards {Cas\,A} ($\sim21$ kJy at 81 MHz, brightest source in northern sky), which is far away from primary field of view. We measure an extremely small ionospheric diffractive scale ($r_{\text{diff}} \approx 430$ m at 60 MHz) towards {Cas\,A} resembling pure Kolmogorov turbulence compared to $r_{\text{diff}} \sim3 - 20$ km towards zenith at 150 MHz for typical ionospheric conditions. This is one of the smallest diffractive scales ever measured at these frequencies. Our work provides insights in understanding the nature of aforementioned effects and mitigating them in future Cosmic Dawn observations (e.g. with SKA-low and HERA) in the same frequency window.Comment: 20 pages, 11 figures, accepted for publication in MNRA