Advances in the study of boundary value problems for nonlinear integrable PDEs

In this review I summarise some of the most significant advances of the last decade in the analysis and solution of boundary value problems for integrable partial differential equations in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions. I focus specifically on a general and widely applicable approach, usually referred to as the Unified Transform or Fokas Transform, that provides a substantial generalisation of the classical Inverse Scattering Transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presents a distinctive and important difference. While the Inverse Scattering Transform follows the "separation of variables" philosophy, albeit in a nonlinear setting, the Unified Transform is a based on the idea of synthesis, rather than separation, of variables. I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalisation to certain nonlinear cases of particular significance.

The elliptic sine-Gordon equation: a nonlinear elliptic integrable PDE

In this paper, we summarise this recent progress to underline the features specific to this nonlinear elliptic case, and we give a new classification of boundary conditions on the semistrip that satisfy a necessary condition for yielding a boundary value problem can be effectively linearised. This classification is based on formulation the equation in terms of an alternative Lax pair.

Nonlinear regional warming with increasing CO₂ concentration

When considering adaptation measures and global climate mitigation goals, stakeholders need regional-scale climate projections, including the range of plausible warming rates. To assist these stakeholders, it is important to understand whether some locations may see disproportionately high or low warming from additional forcing above targets such as 2 K (ref. 1). There is a need to narrow uncertainty2 in this nonlinear warming, which requires understanding how climate changes as forcings increase from medium to high levels. However, quantifying and understanding regional nonlinear processes is challenging. Here we show that regional-scale warming can be strongly superlinear to successive CO2 doublings, using five different climate models. Ensemble-mean warming is superlinear over most land locations. Further, the inter-model spread tends to be amplified at higher forcing levels, as nonlinearities grow—especially when considering changes per kelvin of global warming. Regional nonlinearities in surface warming arise from nonlinearities in global-mean radiative balance, the Atlantic meridional overturning circulation, surface snow/ice cover and evapotranspiration. For robust adaptation and mitigation advice, therefore, potentially avoidable climate change (the difference between business-as-usual and mitigation scenarios) and unavoidable climate change (change under strong mitigation scenarios) may need different analysis methods.

Operational active fire mapping and burnt area identification applicable to Mexican Nature Protection Areas using MODIS and NOAA-AVHRR direct readout data

Since 1999, the National Commission for the Knowledge and Use of the Biodiversity (CONABIO) in Mexico has been developing and managing the “Operational program for the detection of hot-spots using remote sensing techniques”. This program uses images from the MODerate resolution Imaging Spectroradiometer (MODIS) onboard the Terra and Aqua satellites and from the Advanced Very High Resolution Radiometer of the National Oceanic and Atmospheric Administration (NOAA-AVHRR), which are operationally received through the Direct Readout station (DR) at CONABIO. This allows the near-real time monitoring of fire events in Mexico and Central America. In addition to the detection of active fires, the location of hot spots are classified with respect to vegetation types, accessibility, and risk to Nature Protection Areas (NPA). Besides the fast detection of fires, further analysis is necessary due to the considerable effects of forest fires on biodiversity and human life. This fire impact assessment is crucial to support the needs of resource managers and policy makers for adequate fire recovery and restoration actions. CONABIO attempts to meet these requirements, providing post-fire assessment products as part of the management system in particular for satellite-based burnt area mapping. This paper provides an overview of the main components of the operational system and will present an outlook to future activities and system improvements, especially the development of a burnt area product. A special focus will also be placed on the fire occurrence within NPAs of Mexico

Estimating Lorenz's reference state in an ocean with a nonlinear equation of state for seawater

The study of the mechanical energy budget of the oceans using Lorenz available potential energy (APE) theory is based on knowledge of the adiabatically re-arranged Lorenz reference state of minimum potential energy. The compressible and nonlinear character of the equation of state for seawater has been thought to cause the reference state to be ill-defined, casting doubt on the usefulness of APE theory for investigating ocean energetics under realistic conditions. Using a method based on the volume frequency distribution of parcels as a function of temperature and salinity in the context of the seawater Boussinesq approximation, which we illustrate using climatological data, we show that compressibility effects are in fact minor. The reference state can be regarded as a well defined one-dimensional function of depth, which forms a surface in temperature, salinity and density space between the surface and the bottom of the ocean. For a very small proportion of water masses, this surface can be multivalued and water parcels can have up to two statically stable levels in the reference density profile, of which the shallowest is energetically more accessible. Classifying parcels from the surface to the bottom gives a different reference density profile than classifying in the opposite direction. However, this difference is negligible. We show that the reference state obtained by standard sorting methods is equivalent, though computationally more expensive, to the volume frequency distribution approach. The approach we present can be applied systematically and in a computationally efficient manner to investigate the APE budget of the ocean circulation using models or climatological data.

Nonlinear identification using orthogonal forward regression with nested optimal regularization

An efficient data based-modeling algorithm for nonlinear system identification is introduced for radial basis function (RBF) neural networks with the aim of maximizing generalization capability based on the concept of leave-one-out (LOO) cross validation. Each of the RBF kernels has its own kernel width parameter and the basic idea is to optimize the multiple pairs of regularization parameters and kernel widths, each of which is associated with a kernel, one at a time within the orthogonal forward regression (OFR) procedure. Thus, each OFR step consists of one model term selection based on the LOO mean square error (LOOMSE), followed by the optimization of the associated kernel width and regularization parameter, also based on the LOOMSE. Since like our previous state-of-the-art local regularization assisted orthogonal least squares (LROLS) algorithm, the same LOOMSE is adopted for model selection, our proposed new OFR algorithm is also capable of producing a very sparse RBF model with excellent generalization performance. Unlike our previous LROLS algorithm which requires an additional iterative loop to optimize the regularization parameters as well as an additional procedure to optimize the kernel width, the proposed new OFR algorithm optimizes both the kernel widths and regularization parameters within the single OFR procedure, and consequently the required computational complexity is dramatically reduced. Nonlinear system identification examples are included to demonstrate the effectiveness of this new approach in comparison to the well-known approaches of support vector machine and least absolute shrinkage and selection operator as well as the LROLS algorithm.

Nonlinear equalization of Hammerstein OFDM systems

A practical orthogonal frequency-division multiplexing (OFDM) system can generally be modelled by the Hammerstein system that includes the nonlinear distortion effects of the high power amplifier (HPA) at transmitter. In this contribution, we advocate a novel nonlinear equalization scheme for OFDM Hammerstein systems. We model the nonlinear HPA, which represents the static nonlinearity of the OFDM Hammerstein channel, by a B-spline neural network, and we develop a highly effective alternating least squares algorithm for estimating the parameters of the OFDM Hammerstein channel, including channel impulse response coefficients and the parameters of the B-spline model. Moreover, we also use another B-spline neural network to model the inversion of the HPA’s nonlinearity, and the parameters of this inverting B-spline model can easily be estimated using the standard least squares algorithm based on the pseudo training data obtained as a byproduct of the Hammerstein channel identification. Equalization of the OFDM Hammerstein channel can then be accomplished by the usual one-tap linear equalization as well as the inverse B-spline neural network model obtained. The effectiveness of our nonlinear equalization scheme for OFDM Hammerstein channels is demonstrated by simulation results.

Adaptive B-spline neural network based nonlinear equalization for high-order QAM systems with nonlinear transmit high power amplifier

High bandwidth-efficiency quadrature amplitude modulation (QAM) signaling widely adopted in high-rate communication systems suffers from a drawback of high peak-toaverage power ratio, which may cause the nonlinear saturation of the high power amplifier (HPA) at transmitter. Thus, practical high-throughput QAM communication systems exhibit nonlinear and dispersive channel characteristics that must be modeled as a Hammerstein channel. Standard linear equalization becomes inadequate for such Hammerstein communication systems. In this paper, we advocate an adaptive B-Spline neural network based nonlinear equalizer. Specifically, during the training phase, an efficient alternating least squares (LS) scheme is employed to estimate the parameters of the Hammerstein channel, including both the channel impulse response (CIR) coefficients and the parameters of the B-spline neural network that models the HPA’s nonlinearity. In addition, another B-spline neural network is used to model the inversion of the nonlinear HPA, and the parameters of this inverting B-spline model can easily be estimated using the standard LS algorithm based on the pseudo training data obtained as a natural byproduct of the Hammerstein channel identification. Nonlinear equalisation of the Hammerstein channel is then accomplished by the linear equalization based on the estimated CIR as well as the inverse B-spline neural network model. Furthermore, during the data communication phase, the decision-directed LS channel estimation is adopted to track the time-varying CIR. Extensive simulation results demonstrate the effectiveness of our proposed B-Spline neural network based nonlinear equalization scheme.

Identifying, mapping, and monitoring the impact of hybrid firms

There is growing recognition that hybrid organizations can play a critical role in tackling intractable global sustainable development challenges. At the same time, acute social, environmental, and economic challenges are opening up “opportunity” spaces for hybrids. Different institutional contexts are also leading to variable hybrid forms linked to the focus of their mission and their profit-oriented status. This article presents a process for identifying, mapping, and building impact indicators based on a study of 20 hybrid organizations in Sub-Saharan Africa.

Adaptive social protection: mapping the evidence and policy context in the agriculture sector in south Asia

An aim of government and the international community is to respond to global processes and crises through a range of policy and practical approaches that help limit damage from shocks and stresses. Three approaches to vulnerability reduction that have become particularly prominent in recent years are social protection (SP), disaster risk reduction (DRR) and climate change adaptation (CCA). Although these approaches have much in common, they have developed separately over the last two decades. However, given the increasingly complex and interlinked array of risks that poor and vulnerable people face, it is likely that they will not be sufficient in the long run if they continue to be applied in isolation from one another. In recognition of this challenge, the concept of Adaptive Social Protection (ASP) has been developed. ASP refers to a series of measures which aims to build resilience of the poorest and most vulnerable people to climate change by combining elements of SP, DRR and CCA in programmes and projects. The aim of this paper is to provide an initial assessment of the ways in which these elements are being brought together in development policy and practice. It does this by conducting a meta-analysis of 124 agricultural programmes implemented in five countries in south Asia. These are Afghanistan, Bangladesh, India, Nepal and Pakistan. The findings show that full integration of SP, DRR and CCA is relatively limited in south Asia, although there has been significant progress in combining SP and DRR in the last ten years. Projects that combine elements of SP, DRR and CCA tend to emphasise broad poverty and vulnerability reduction goals relative to those that do not. Such approaches can provide valuable lessons and insights for the promotion of climate resilient livelihoods amongst policymakers and practitioners.

Genome-wide mapping of 5-hydroxymethylcytosine in three rice cultivars reveals its preferential localization in transcriptionally silent transposable element genes

5-Hydroxymethylcytosine (5hmC), a modified form of cytosine that is considered the sixth nucleobase in DNA, has been detected in mammals and is believed to play an important role in gene regulation. In this study, 5hmC modification was detected in rice by employing a dot-blot assay, and its levels was further quantified in DNA from different rice tissues using liquid chromatography-multistage mass spectrometry (LC-MS/MS/MS). The results showed large intertissue variation in 5hmC levels. The genome-wide profiles of 5hmC modification in three different rice cultivars were also obtained using a sensitive chemical labelling followed by a next-generation sequencing method. Thousands of 5hmC peaks were identified, and a comparison of the distributions of 5hmC among different rice cultivars revealed the specificity and conservation of 5hmC modification. The identified 5hmC peaks were significantly enriched in heterochromatin regions,and mainly located in transposable element (TE) genes, especially around retrotransposons. The correlation analysis of 5hmC and gene expression data revealed a close association between 5hmC and silent TEs. These findings provide a resource for plant DNA 5hmC epigenetic studies and expand our knowledge of 5hmC modification.

Mapping power law distributions in digital health social networks: methods, interpretations, and practical implications

BACKGROUND: Social networks are common in digital health. A new stream of research is beginning to investigate the mechanisms of digital health social networks (DHSNs), how they are structured, how they function, and how their growth can be nurtured and managed. DHSNs increase in value when additional content is added, and the structure of networks may resemble the characteristics of power laws. Power laws are contrary to traditional Gaussian averages in that they demonstrate correlated phenomena. OBJECTIVES: The objective of this study is to investigate whether the distribution frequency in four DHSNs can be characterized as following a power law. A second objective is to describe the method used to determine the comparison. METHODS: Data from four DHSNs—Alcohol Help Center (AHC), Depression Center (DC), Panic Center (PC), and Stop Smoking Center (SSC)—were compared to power law distributions. To assist future researchers and managers, the 5-step methodology used to analyze and compare datasets is described. RESULTS: All four DHSNs were found to have right-skewed distributions, indicating the data were not normally distributed. When power trend lines were added to each frequency distribution, R(2) values indicated that, to a very high degree, the variance in post frequencies can be explained by actor rank (AHC .962, DC .975, PC .969, SSC .95). Spearman correlations provided further indication of the strength and statistical significance of the relationship (AHC .987. DC .967, PC .983, SSC .993, P<.001). CONCLUSIONS: This is the first study to investigate power distributions across multiple DHSNs, each addressing a unique condition. Results indicate that despite vast differences in theme, content, and length of existence, DHSNs follow properties of power laws. The structure of DHSNs is important as it gives insight to researchers and managers into the nature and mechanisms of network functionality. The 5-step process undertaken to compare actor contribution patterns can be replicated in networks that are managed by other organizations, and we conjecture that patterns observed in this study could be found in other DHSNs. Future research should analyze network growth over time and examine the characteristics and survival rates of superusers.

High Royds: an integrated, analytical approach for mapping the unmarked burials of a pauper cemetery

Applying geophysical techniques to detect and map the physical extent of individual unmarked graves proves difficult in many cases. The success of individual geophysical techniques for detecting unmarked graves may be due to a poor understanding of the nature of the graves themselves, the context in which they lie in, and temporal changes to the burial state. Given the unpredictability of these variables, it is surprising that grave prospection is often undertaken using only a single method. This paper presents a multi-methodological survey strategy for detecting unmarked burials and utilises an analytical approach for visualising and evaluating survey results.