Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

The renormalization group (RG) method is one of the singular perturbation methods which is used in search for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, inheritance of symmetries from those for the original equation to those for the RG equation, are proved. Further it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper-) normal forms theory, the center manifold reduction, the geometric singular perturbation method and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.Comment: publised as SIAM j. on Appl. Dyn.Syst., Vol.8, 1066-1115 (2009

A matrix solution to pentagon equation with anticommuting variables

We construct a solution to pentagon equation with anticommuting variables living on two-dimensional faces of tetrahedra. In this solution, matrix coordinates are ascribed to tetrahedron vertices. As matrix multiplication is noncommutative, this provides a "more quantum" topological field theory than in our previous works

A SOM-based analysis of the drivers of the 2015–2017 Western Cape drought in South Africa

The multi-year (2015–2017) drought in the South West of the Western Cape (SWC) caused a severe water shortage in the summer of 2017–2018, with damaging impacts on the local and regional economy, and Cape Town being in the news one of the first major cities to potentially run out of water. Here, we assess the links between the rainfall deficits during the drought and (a) large scale circulation patterns, (b) moisture transport, and (c) convective available potential energy (CAPE). We used self-organising maps (SOM) analysis to classify daily ERA-interim 850 hPa geopotential height for the period 1979–2017 (March–October) into synoptic types. This allowed us to identify the dominant synoptic states over Southern Africa that influence the local climate in the area affected by the drought. The results show that (a) the frequency of nodes with rain-bearing circulation types decreased during the drought; (b) the amount of rain falling on days that did have rain-bearing circulation types was reduced, especially in the shoulder seasons (March–May and August–October); (c) the rainfall reduction was also associated with anomalously low moisture transport, and convective energy (CAPE), over SWC. These results add to the existing knowledge of drivers of the Cape Town drought, providing an understanding of underlying synoptic processes

Cloud Condensation Nuclei properties of model and atmospheric HULIS

Humic like substances (HULIS) have been identified as a major fraction of the organic component of atmospheric aerosols. These large multifunctional compounds of both primary and secondary sources are surface active and water soluble. Hence, it is expected that they could affect activation of organic aerosols into cloud droplets. We have compared the activation of aerosols containing atmospheric HULIS extracted from fresh, aged and pollution particles to activation of size fractionated fulvic acid from an aquatic source (Suwannee River Fulvic Acid), and correlated it to the estimated molecular weight and measured surface tension. A correlation was found between CCN-activation diameter of SRFA fractions and number average molecular weight of the fraction. The lower molecular weight fractions activated at lower critical diameters, which is explained by the greater number of solute species in the droplet with decreasing molecular weight. The three aerosol-extracted HULIS samples activated at lower diameters than any of the size-fractionated or bulk SRFA. The Köhler model was found to account for activation diameters, provided that accurate physico-chemical parameters are known

Sensing and Mining Urban Qualities in Smart Cities

The emergence of the Internet of Things in Smart Cities questions how the future citizens will perceive their predominant living and working environments and what quality of living they can experience within it, for instance the level of everyday stress. However, perception and experienced stress levels are challenging metrics to measure and are even more challenging to correlate with an underlying causal-effectual relationship in such stimulus abundant environments. The Internet of Things, enabled by several pervasive and ubiquitous devices such as smart phones and smart sensors, can provide real-time contextual information that can be used by advanced data science methodologies to generate new insights about urban qualities in Smart Cities and how they can be improved. The goal of this study is to show the predominant factors, which influence perceptual qualities of inhabitants in a Smart City equipped with sensing capabilities by the Internet of Things. To serve this goal, a novel data collection process for Smart Cities is introduced that involves (i) environmental data, such noise, dust, illuminance, temperature, relative humidity, (ii) location/mobility data, such as GNSS and citizens density detected via WiFi, and (iii) perceptual social data collected by citizens' responses in smart phones. These fine-grained real-time data can provide invaluable insights about the spatial correlations of the sensor measurements as well as the spatial and citizens' similarity illustrated. The data analysis illustrated reveals significant links between stress level and environmental changes observed