The foundations of spectral computations via the Solvability Complexity Index hierarchy: Part I

The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. Recent progress and the current paper reveal that, unlike the finite-dimensional case, infinite-dimensional problems yield a highly intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithms. Classifying spectral problems and providing optimal algorithms is uncharted territory in the foundations of computational mathematics. This paper is the first of a two-part series establishing the foundations of computational spectral theory through the Solvability Complexity Index (SCI) hierarchy and has three purposes. First, we establish answers to many longstanding open questions on the existence of algorithms. We show that for large classes of partial differential operators on unbounded domains, spectra can be computed with error control from point sampling operator coefficients. Further results include computing spectra of operators on graphs with error control, the spectral gap problem, spectral classifications, and discrete spectra, multiplicities and eigenspaces. Second, these classifications determine which types of problems can be used in computer-assisted proofs. The theory for this is virtually non-existent, and we provide some of the first results in this infinite classification theory. Third, our proofs are constructive, yielding a library of new algorithms and techniques that handle problems that before were out of reach. We show several examples on contemporary problems in the physical sciences. Our approach is closely related to Smale's program on the foundations of computational mathematics initiated in the 1980s, as many spectral problems can only be computed via several limits, a phenomenon shared with the foundations of polynomial root finding with rational maps, as proved by McMullen

How to Compute Spectra with Error Control.

Computing the spectra of operators is a fundamental problem in the sciences, with wide-ranging applications in condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that in certain cases converge to the spectrum, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) provides approximate eigenvectors. This may lead to incorrect simulations. It has been an open problem since the 1950s to decide whether such reliable methods exist at all. We affirmatively resolve this question, and the algorithms provided are optimal, realizing the boundary of what digital computers can achieve. Moreover, they are easy to implement and parallelize, offer fundamental speed-ups, and allow problems that before, regardless of computing power, were out of reach. Results are demonstrated on difficult problems such as the spectra of quasicrystals and non-Hermitian phase transitions in optics.This work was supported by Engineering and Physical Sciences Research Council Grants No. EP/L016516/1, No. EP/R008272/1, No. EP/N014588/1, and No. EP/ L003457/1, as well as a Royal Society University Research Fellowship

Securing tropical forest carbon: the contribution of protected areas to REDD

Forest loss and degradation in the tropics contribute 6-17% of all greenhouse gas emissions. Protected areas cover 217.2 million ha (19.6%) of the world's humid tropical forests and contain c. 70.3 petagrams of carbon (Pg C) in biomass and soil to 1 m depth. Between 2000 and 2005, we estimate that 1.75 million ha of forest were lost from protected areas in humid tropical forests, causing the emission of 0.25-0.33 Pg C. Protected areas lost about half as much carbon as the same area of unprotected forest. We estimate that the reduction of these carbon emissions from ongoing deforestation in protected sites in humid tropical forests could be valued at USD 6,200-7,400 million depending on the land use after clearance. This is >1.5 times the estimated spending on protected area management in these regions. Improving management of protected areas to retain forest cover better may be an important, although certainly not sufficient, component of an overall strategy for reducing emissions from deforestation and forest degradation (REDD

First-Pass Meconium Samples from Healthy Term Vaginally-Delivered Neonates : An Analysis of the Microbiota

Acknowledgments The authors would like to thank the parents who consented to provide samples with limited notice at an emotional and stressful time. This work was supported entirely from personal donations to the neonatal endowments fund at Aberdeen Maternity Hospital and we thank families for their continued generosity, year-on-year. The Rowett Institute of Nutrition and Health receives funding from the Scottish Government (SG-RESAS). Funding: This work was funded from NHS Grampian Neonatal Endowments. The Rowett Institute receives funding from the Rural and Environmental Science and Analytical Services programme of the Scottish Government. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Peer reviewedPublisher PD

Multiple aneurysms in childhood

AbstractArterial aneurysms in children are rare. When present, they are often associated with connective tissue disorders or arteritidies. Idiopathic aneurysms occurring at multiple sites throughout the arterial tree are rare, with only ten cases reported. This report describes a case of multiple arterial aneurysms of uncertain origin involving upper-extremity, extracranial cerebrovascular, aortoiliac, and renal arteries in a 14-year-old boy. The clinical presentation, vascular reconstruction, pathologic findings, and a brief review of the literature are described

An Initial Report of Sleep Disorders in Women in the U.S. Military

The article of record as published may be found at http://dx.doi.org/10.1093/milmed/usx116Sleep disorders are increasingly recognized in active duty service members (ADSM). While there are multiple studies in male ADSM, there are limited data regarding sleep disorders in women in the military. The purpose of this study was to characterize sleep disorders in female ADSM referred for clinical evaluation to provide a better understanding of this unique population. Materials and Methods: We conducted a retrospective review of female ADSM who underwent a sleep medicine evaluation and an attended polysomnogram (PSG). Demographic and polysomnogram variables, as well as medical records, were reviewed. Associated illnesses to include post-traumatic stress disorder, pain disorders, anxiety, and depression, were recorded. Results: The cohort consisted of 101 women. The average age was 33.9 ± 9.0 years and body mass index was 27.3 ± 4.5, with an average Epworth Sleepiness Scale score of 12.9 ± 5.2, and Insomnia Severity Index score of 17.6 ± 5.7. Overall, 36.6% were diagnosed with insomnia only, 14.9% with obstructive sleep apnea (OSA) only, and 34.7% met diagnostic criteria for both insomnia and OSA. The average apnea-hypopnea index for the entire cohort was 5.37 ± 7.04/h whereas it was 10.34 ± 3.14/h for those meeting diagnostic criteria for OSA. The women referred for sleep evaluations had the following rates of associated illnesses: pain disorders (59.4%), anxiety (48.5%), depression (46.5%), and post-traumatic stress disorder (21.8%). For patients with OSA, the relative risk of having post-traumatic stress disorder was 2.72 (95% confidence interval 1.16–6.39). Conclusions: Women in the U.S. military who have sleep disorders have a high rate of behavioral medicine and pain disorders. Interestingly, nearly 50% of active duty females referred for a sleep study have OSA while not necessarily manifesting the typical signs of obesity or increased age. The reasons for this finding are not completely understood, though factors related to military service may potentially contribute. The findings from our study indicate a need for increased awareness and evaluation of sleep disorders in women in the military, especially those with behavioral medicine disorders

On the infinite-dimensional QR algorithm

Abstract: Spectral computations of infinite-dimensional operators are notoriously difficult, yet ubiquitous in the sciences. Indeed, despite more than half a century of research, it is still unknown which classes of operators allow for the computation of spectra and eigenvectors with convergence rates and error control. Recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation, and no convergence rates nor error control is possible. This begs the question: which classes of operators allow for computations with convergence rates and error control? In this paper, we address this basic question, and the algorithm used is an infinite-dimensional version of the QR algorithm. Indeed, we generalise the QR algorithm to infinite-dimensional operators. We prove that not only is the algorithm executable on a finite machine, but one can also recover the extremal parts of the spectrum and corresponding eigenvectors, with convergence rates and error control. This allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture. The algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches (that are notorious for providing false solutions). We also find that in some cases the IQR algorithm performs better than predicted by theory and make conjectures for future study

Land‐cover and land‐use change trajectory hopping facilitates estate‐crop expansion into protected forests in Indonesia

Protected areas (PAs) have been regarded as a critical strategy to protect natural forest (NF) and biodiversity. Estate‐crop expansion is an important driver of deforestation in Indonesia. Yet, little is known regarding the temporal dynamics of PA effectiveness in preventing estate‐crop expansion into NF. We employ Cox proportional hazard models and their extensions to characterize the dynamics of estate‐crop expansion into NF in Indonesia during 1996–2015. The results show that PA effectiveness in Sumatra decreased over time and became insignificant in 2012–2015. A multistate modeling analysis shows that hopping in land‐cover and land‐use change (LCLUC) trajectories with shrub and/or bare ground as intermediates has decreased PA effectiveness and facilitated the expansion. Preventing LCLUC trajectory hopping becomes crucial to biodiversity conservation because it tends to occur at lowland forest, diminishing natural habitat area and increasing NF isolation