Philosophical transactions. Series A, Mathematical, physical, and engineering sciences

Independent component analysis is a probabilistic method for learning a linear transform of a random vector. The goal is to find components that are maximally independent and non-Gaussian (non-normal). Its fundamental difference to classical multi-variate statistical methods is in the assumption of non-Gaussianity, which enables the identification of original, underlying components, in contrast to classical methods. The basic theory of independent component analysis was mainly developed in the 1990s and summarized, for example, in our monograph in 2001. Here, we provide an overview of some recent developments in the theory since the year 2000. The main topics are: analysis of causal relations, testing independent components, analysing multiple datasets (three-way data), modelling dependencies between the components and improved methods for estimating the basic model.

Access to a safe supply of water is a human right. However, with growing populations, global warming and contamination due to human activity, it is one that is increasingly under threat. It is hoped that nature can inspire the creation of materials to aid in the supply and management of water, from water collection and purification to water source clean-up and rehabilitation from oil contamination. Many species thrive in even the driest places, with some surviving on water harvested from fog. By studying these species, new materials can be developed to provide a source of fresh water from fog for communities across the globe. The vast majority of water on the Earth is in the oceans. However, current desalination processes are energy-intensive. Systems in our own bodies have evolved to transport water efficiently while blocking other molecules and ions. Inspiration can be taken from such to improve the efficiency of desalination and help purify water containing other contaminants. Finally, oil contamination of water from spills or the fracking technique can be a devastating environmental disaster. By studying how natural surfaces interact with liquids, new techniques can be developed to clean up oil spills and further protect our most precious resource.

This article is part of the themed issue ‘Bioinspired hierarchically structured surfaces for green science’.

Plastics have transformed our modern world. With a range of outstanding properties, they are used in an ever-widening range of applications. However, the linear economy of their use means that a large volume of plastics is discarded after use. It is believed that approximately 80% of the estimated total 6.3 Bt of plastics ever produced have been discarded, representing not only a huge loss of valuable resources, but mismanaged waste is also the origin of an ever-increasing environmental disaster. Strategies to prevent loss of materials resources and damage to the environment are elements of a circular plastics economy that aims to maintain plastics at their highest value for the longest time possible and at the same time improve the economy and prevent detrimental environmental impact. The latter in particular is driving recent changes in policies and legislation across the world that are rapidly being introduced in order to solve these environmental issues. The achievement of a circular economy will require not only innovative technical developments, but also major economic investment and changes to business practice coupled with significant changes in social behaviour. This paper summarizes the complex and highly interrelated technical issues and provides an overview of the major challenges, potential solutions and opportunities required to achieve and operate a circular plastics economy.

This article is part of a discussion meeting issue ‘Science to enable the circular economy’.

A family of Cu(II)-based metal–organic frameworks (MOFs) has been synthesized using three pyridyl-isophthalate ligands, H ₂ L ¹ (4′-(pyridin-4-yl)biphenyl-3,5-dicarboxylic acid), H ₂ L ² (4′′-(pyridin-4-yl)-1,1′:4′,1′′-terphenyl-3,5-dicarboxylic acid) and H ₂ L ³ (5-[4-(pyridin-4-yl)naphthalen-1-yl]benzene-1,3-dicarboxylic acid). Although in each case the pyridyl-isophthalate ligands adopt the same pseudo-octahedral [Cu ₂ (O ₂ CR) ₄ N ₂ ] paddlewheel coordination modes, the resulting frameworks are structurally diverse, particularly in the case of the complex of Cu(II) with H ₂ L ³ , which leads to three distinct supramolecular isomers, each derived from Kagomé and square nets. In contrast to [Cu(L ² )] and the isomers of [Cu(L ³ )], [Cu(L ¹ )] exhibits permanent porosity. Thus, the gas adsorption properties of [Cu(L ¹ )] were investigated with N ₂ , CO ₂ and H ₂ , and the material exhibits an isosteric heat of adsorption competitive with leading MOF sorbents for CO ₂ . [Cu(L ¹ )] displays high H ₂ adsorption, with the density in the pores approaching that of liquid H ₂ .

This article is part of the themed issue ‘Coordination polymers and metal–organic frameworks: materials by design’.

Evolving constituent composition and organization are important determinants of the biomechanical behaviour of soft tissues. In arteries, vascular smooth muscle cells and fibroblasts continually produce and degrade matrix constituents in preferred modes and at altered rates in response to changing mechanical stimuli. Smooth muscle cells similarly exhibit vasoactive changes that contribute to the control of overall structure, function and mechanical behaviour. Constrained mixture models provide a useful framework in which to quantify arterial growth and remodelling for they can account for cell-mediated changes in individual structurally significant constituents. Our simulations show that the combined effects of changing mass density turnover and vasoactivity, as well as the prestretch at which constituents are incorporated within extant matrix, are essential to capture salient features of bounded arterial growth and remodelling. These findings emphasize the importance of formulating biologically motivated constitutive relations in any theory of growth and remodelling and distinct advantages of the constrained mixture approach, in particular.

Water vapour continuum absorption is an important contributor to the Earth's radiative cooling and energy balance. Here, we describe the development and status of the MT_CKD (MlawerTobinCloughKneizysDavies) water vapour continuum absorption model. The perspective adopted in developing the MT_CKD model has been to constrain the model so that it is consistent with quality analyses of spectral atmospheric and laboratory measurements of the foreign and self continuum. For field measurements, only cases for which the characterization of the atmospheric state has been highly scrutinized have been used. Continuum coefficients in spectral regions that have not been subject to compelling analyses are determined by a mathematical formulation of the spectral shape associated with each water vapour monomer line. This formulation, which is based on continuum values in spectral regions in which the coefficients are well constrained by measurements, is applied consistently to all water vapour monomer lines from the microwave to the visible. The results are summed-up (separately for the foreign and self) to obtain continuum coefficients from 0 to 20 000 cm ⁻¹ . For each water vapour line, the MT_CKD line shape formulation consists of two components: exponentially decaying far wings of the line plus a contribution from a water vapour molecule undergoing a weak interaction with a second molecule. In the MT_CKD model, the first component is the primary agent for the continuum between water vapour bands, while the second component is responsible for the majority of the continuum within water vapour bands. The MT_CKD model should be regarded as a semi-empirical model with strong constraints provided by the known physics. Keeping the MT_CKD continuum consistent with current observational studies necessitates periodic updates to the water vapour continuum coefficients. In addition to providing details on the MT_CKD line shape formulation, we describe the most recent update to the model, MT_CKD_2.5, which is based on an analysis of satellite- and ground-based observations from 2385 to 2600 cm ⁻¹ (approx. 4 μm).