Fractal spatialities

This paper argues for the use of ‘fractals’ in theorising sociospatial relations. From a realist position, a nonmathematical but nonmetaphoric and descriptive view of ‘fractals’ is advanced. Insights from the natural sciences are combined with insights on the position of the observer from Luhmann and notions of assemblages and repetitions from Deleuze. It is argued that the notion of ‘fractals’ can augment current understanding of sociospatialities in three ways. First, it can pose questions about the scalar position of the observer or the grain of observation; second, as a signifier of particular attributes, it prompts observation and description of particular structuring processes; and third, the epistemic access afforded by the concept can open up possibilities for transformative interventions and thereby inform the same. The theoretical usefulness of the concept is demonstrated by discussing the territory, place, scale, and networks (TPSN) model for theorising sociospatial relations advanced by B Jessop, N Brenner, and M Jones in their 2008 paper “Theorizing sociospatial relations”, published in this journal (volume 26, pages 389–401). It is suggested that a heuristic arising from a ‘fractal’ ontology can contribute to a polymorphous, as opposed to polyvalent, understanding of sociospatial relations.

Fractal spaces in planning and governance

This paper discusses concepts of space within the planning literature, the issues they give rise to and the gaps they reveal. It then introduces the notion of 'fractals' borrowed from complexity theory and illustrates how it unconsciously appears in planning practice. It then moves on to abstract the core dynamics through which fractals can be consciously applied and illustrates their working through a reinterpretation of the People's Planning Campaign of Kerala, India. Finally it highlights the key contribution of the fractal concept and the advantages that this conceptualisation brings to planning.

Solution-free sets for sums of binary forms

In this paper, we obtain quantitative estimates for the asymptotic density of subsets of the integer lattice Z2 that contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov’s mean value theorem applicable to binary forms.

Wave activity in the tropical tropopause layer in seven reanalysis and four chemistry climate model data sets

Sub-seasonal variability including equatorial waves significantly influence the dehydration and transport processes in the tropical tropopause layer (TTL). This study investigates the wave activity in the TTL in 7 reanalysis data sets (RAs; NCEP1, NCEP2, ERA40, ERA-Interim, JRA25, MERRA, and CFSR) and 4 chemistry climate models (CCMs; CCSRNIES, CMAM, MRI, and WACCM) using the zonal wave number-frequency spectral analysis method with equatorially symmetric-antisymmetric decomposition. Analyses are made for temperature and horizontal winds at 100 hPa in the RAs and CCMs and for outgoing longwave radiation (OLR), which is a proxy for convective activity that generates tropopause-level disturbances, in satellite data and the CCMs. Particular focus is placed on equatorial Kelvin waves, mixed Rossby-gravity (MRG) waves, and the Madden-Julian Oscillation (MJO). The wave activity is defined as the variance, i.e., the power spectral density integrated in a particular zonal wave number-frequency region. It is found that the TTL wave activities show significant difference among the RAs, ranging from ∼0.7 (for NCEP1 and NCEP2) to ∼1.4 (for ERA-Interim, MERRA, and CFSR) with respect to the averages from the RAs. The TTL activities in the CCMs lie generally within the range of those in the RAs, with a few exceptions. However, the spectral features in OLR for all the CCMs are very different from those in the observations, and the OLR wave activities are too low for CCSRNIES, CMAM, and MRI. It is concluded that the broad range of wave activity found in the different RAs decreases our confidence in their validity and in particular their value for validation of CCM performance in the TTL, thereby limiting our quantitative understanding of the dehydration and transport processes in the TTL.

Evidence for changes in stratospheric transport and mixing over the past three decades based on multiple data sets and tropical leaky pipe analysis

Variability in the strength of the stratospheric Lagrangian mean meridional or Brewer-Dobson circulation and horizontal mixing into the tropics over the past three decades are examined using observations of stratospheric mean age of air and ozone. We use a simple representation of the stratosphere, the tropical leaky pipe (TLP) model, guided by mean meridional circulation and horizontal mixing changes in several reanalyses data sets and chemistry climate model (CCM) simulations, to help elucidate reasons for the observed changes in stratospheric mean age and ozone. We find that the TLP model is able to accurately simulate multiyear variability in ozone following recent major volcanic eruptions and the early 2000s sea surface temperature changes, as well as the lasting impact on mean age of relatively short-term circulation perturbations. We also find that the best quantitative agreement with the observed mean age and ozone trends over the past three decades is found assuming a small strengthening of the mean circulation in the lower stratosphere, a moderate weakening of the mean circulation in the middle and upper stratosphere, and a moderate increase in the horizontal mixing into the tropics. The mean age trends are strongly sensitive to trends in the horizontal mixing into the tropics, and the uncertainty in the mixing trends causes uncertainty in the mean circulation trends. Comparisons of the mean circulation and mixing changes suggested by the measurements with those from a recent suite of CCM runs reveal significant differences that may have important implications on the accurate simulation of future stratospheric climate.

The accuracy of SST retrievals from AATSR: An initial assessment through geophysical validation against in situ radiometers, buoys and other SST data sets

The Advanced Along-Track Scanning Radiometer (AATSR) was launched on Envisat in March 2002. The AATSR instrument is designed to retrieve precise and accurate global sea surface temperature (SST) that, combined with the large data set collected from its predecessors, ATSR and ATSR-2, will provide a long term record of SST data that is greater than 15 years. This record can be used for independent monitoring and detection of climate change. The AATSR validation programme has successfully completed its initial phase. The programme involves validation of the AATSR derived SST values using in situ radiometers, in situ buoys and global SST fields from other data sets. The results of the initial programme presented here will demonstrate that the AATSR instrument is currently close to meeting its scientific objectives of determining global SST to an accuracy of 0.3 K (one sigma). For night time data, the analysis gives a warm bias of between +0.04 K (0.28 K) for buoys to +0.06 K (0.20 K) for radiometers, with slightly higher errors observed for day time data, showing warm biases of between +0.02 (0.39 K) for buoys to +0.11 K (0.33 K) for radiometers. They show that the ATSR series of instruments continues to be the world leader in delivering accurate space-based observations of SST, which is a key climate parameter.

Complex extreme learning machine applications in terahertz pulsed signals feature sets

This paper presents a novel approach to the automatic classification of very large data sets composed of terahertz pulse transient signals, highlighting their potential use in biochemical, biomedical, pharmaceutical and security applications. Two different types of THz spectra are considered in the classification process. Firstly a binary classification study of poly-A and poly-C ribonucleic acid samples is performed. This is then contrasted with a difficult multi-class classification problem of spectra from six different powder samples that although have fairly indistinguishable features in the optical spectrum, they also possess a few discernable spectral features in the terahertz part of the spectrum. Classification is performed using a complex-valued extreme learning machine algorithm that takes into account features in both the amplitude as well as the phase of the recorded spectra. Classification speed and accuracy are contrasted with that achieved using a support vector machine classifier. The study systematically compares the classifier performance achieved after adopting different Gaussian kernels when separating amplitude and phase signatures. The two signatures are presented as feature vectors for both training and testing purposes. The study confirms the utility of complex-valued extreme learning machine algorithms for classification of the very large data sets generated with current terahertz imaging spectrometers. The classifier can take into consideration heterogeneous layers within an object as would be required within a tomographic setting and is sufficiently robust to detect patterns hidden inside noisy terahertz data sets. The proposed study opens up the opportunity for the establishment of complex-valued extreme learning machine algorithms as new chemometric tools that will assist the wider proliferation of terahertz sensing technology for chemical sensing, quality control, security screening and clinic diagnosis. Furthermore, the proposed algorithm should also be very useful in other applications requiring the classification of very large datasets.

Fractal-to-Euclidean crossover of the isotropy restoration feature in a family of fractal resistor networks

Fractal with microscopic anisotropy shows a unique type of macroscopic isotropy restoration phenomenon that is absent in Euclidean space [M. T. Barlow et al., Phys. Rev. Lett. 75, 3042]. In this paper the isotropy restoration feature is considered for a family of two-dimensional Sierpinski gasket type fractal resistor networks. A parameter xi is introduced to describe this phenomenon. Our numerical results show that xi satisfies the scaling law xi similar to l(-alpha), where l is the system size and alpha is an exponent independent of the degree of microscopic anisotropy, characterizing the isotropy restoration feature of the fractal systems. By changing the underlying fractal structure towards the Euclidean triangular lattice through increasing the side length b of the gasket generators, the fractal-to-Euclidean crossover behavior of the isotropy restoration feature is discussed.

Weighted alpha-rate dominating sets in social networks

We are looking into variants of a domination set problem in social networks. While randomised algorithms for solving the minimum weighted domination set problem and the minimum alpha and alpha-rate domination problem on simple graphs are already present in the literature, we propose here a randomised algorithm for the minimum weighted alpha-rate domination set problem which is, to the best of our knowledge, the first such algorithm. A theoretical approximation bound based on a simple randomised rounding technique is given. The algorithm is implemented in Python and applied to a UK Twitter mentions networks using a measure of individuals’ influence (klout) as weights. We argue that the weights of vertices could be interpreted as the costs of getting those individuals on board for a campaign or a behaviour change intervention. The minimum weighted alpha-rate dominating set problem can therefore be seen as finding a set that minimises the total cost and each individual in a network has at least alpha percentage of its neighbours in the chosen set. We also test our algorithm on generated graphs with several thousand vertices and edges. Our results on this real-life Twitter networks and generated graphs show that the implementation is reasonably efficient and thus can be used for real-life applications when creating social network based interventions, designing social media campaigns and potentially improving users’ social media experience.

The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.

Fractal geometry of aggregate snowflakes revealed by triple wavelength radar measurements

Radar reflectivity measurements from three different wavelengths are used to retrieve information about the shape of aggregate snowflakes in deep stratiform ice clouds. Dual-wavelength ratios are calculated for different shape models and compared to observations at 3, 35 and 94 GHz. It is demonstrated that many scattering models, including spherical and spheroidal models, do not adequately describe the aggregate snowflakes that are observed. The observations are consistent with fractal aggregate geometries generated by a physically-based aggregation model. It is demonstrated that the fractal dimension of large aggregates can be inferred directly from the radar data. Fractal dimensions close to 2 are retrieved, consistent with previous theoretical models and in-situ observations.

Fractal geometry and architecture design: case study review

The idea of buildings in harmony with nature can be traced back to ancient times. The increasing concerns on sustainability oriented buildings have added new challenges in building architectural design and called for new design responses. Sustainable design integrates and balances the human geometries and the natural ones. As the language of nature, it is, therefore, natural to assume that fractal geometry could play a role in developing new forms of aesthetics and sustainable architectural design. This paper gives a brief description of fractal geometry theory and presents its current status and recent developments through illustrative review of some fractal case studies in architecture design, which provides a bridge between fractal geometry and architecture design.

A method for merging nadir-sounding climate records, with an application to the global-mean stratospheric temperature data sets from SSU and AMSU

A method is proposed for merging different nadir-sounding climate data records using measurements from high-resolution limb sounders to provide a transfer function between the different nadir measurements. The two nadir-sounding records need not be overlapping so long as the limb-sounding record bridges between them. The method is applied to global-mean stratospheric temperatures from the NOAA Climate Data Records based on the Stratospheric Sounding Unit (SSU) and the Advanced Microwave Sounding Unit-A (AMSU), extending the SSU record forward in time to yield a continuous data set from 1979 to present, and providing a simple framework for extending the SSU record into the future using AMSU. SSU and AMSU are bridged using temperature measurements from the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS), which is of high enough vertical resolution to accurately represent the weighting functions of both SSU and AMSU. For this application, a purely statistical approach is not viable since the different nadir channels are not sufficiently linearly independent, statistically speaking. The near-global-mean linear temperature trends for extended SSU for 1980–2012 are −0.63 ± 0.13, −0.71 ± 0.15 and −0.80 ± 0.17 K decade−1 (95 % confidence) for channels 1, 2 and 3, respectively. The extended SSU temperature changes are in good agreement with those from the Microwave Limb Sounder (MLS) on the Aura satellite, with both exhibiting a cooling trend of ~ 0.6 ± 0.3 K decade−1 in the upper stratosphere from 2004 to 2012. The extended SSU record is found to be in agreement with high-top coupled atmosphere–ocean models over the 1980–2012 period, including the continued cooling over the first decade of the 21st century.

A multifractal zeta function for Gibbs measures supported on cookie-cutter sets

Starting with the work of Lapidus and van Frankenhuysen a number of papers have introduced zeta functions as a way of capturing multifractal information. In this paper we propose a new multifractal zeta function and show that under certain conditions the abscissa of convergence yields the Hausdorff multifractal spectrum for a class of measures.