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Aim: The process of urbanization can lead to specialist species being replaced by generalist species in space and time, increasing similarity among bird communities. This phenomenon is termed biotic homogenization and is directly related to taxonomic and functional diversity. However, the effects of urbanization on phylogenetic diversity remain unclear. Our study addresses the effects of the process of urbanization on the evolutionary distinctiveness (a quantitative measure of the genetic or evolutionary uniqueness of species) of bird communities. Location: Europe. Methods: Mixed models were used to compare the effects of urbanization on the evolutionary distinctiveness of bird communities in rural and urban environments in six different European cities from different ecoregions. Results: Our study presents unique large-scale evidence of a negative impact of urban environments on the evolutionary uniqueness of birds. Compared with bird communities in rural environments, bird communities in urban environments have lower average evolutionary distinctiveness in all countries, independent of ecoregion, and these values are unrelated to the taxonomic diversity present in each country. Main conclusions: Our findings provide important information on the spectrum of effects on global biodiversity of changes in land use related to the process of urbanization. Therefore, urban environments are a factor of concern for maintaining diversity across the tree of life of birds, and we suggest that urbanization planning could help buffer against extreme loss of phylogenetic diversity caused by this process.

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations: The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that

The present thesis addresses a broad range of weak convergence problems arising in Nonlinear Analysis. The thesis is divided into three related but essentially independent parts. In the first part we study the general theory of Compensated Compactness. We begin by giving a new characterization of partial differential operators with constant rank, a mild non-degeneracy assumption which plays an important role in the theory. We then characterize completely the class of nonlinearities which are weakly continuous with respect to constant rank PDEs; in particular, we prove that it agrees with the class of nonlinear operators with Hardy space integrability, answering positively a question by Coifman-Lions-Meyer-Semmes. As an application of this theory we study homogenization problems, both with and without constant rank assumptions. In the constant rank setting we revisit the classical G-closure problem and discuss its connection with quasiconvexity. In the non-constant rank setting we study a homogenization problem for the Einstein vacuum equations in General Relativity: under some symmetry and gauge assumptions, we prove a conjecture by Burnett from 1989 which describes the effective behaviour of a sequence of vacuum space-times. This part of the thesis contains joint work with Jan Kristensen (University of Oxford), Bogdan Raită (MPI Leipzig), Matthew Schrecker (UCL) and Rita Teixeira da Costa (University of Cambridge). The second part of this thesis is concerned with quasiconvexity in the classical, curl-free, Calculus of Variations. We contribute to the understanding of the geometry of the class of quasiconvex functions, in particular through its extremal points. We prove a Choquet-type theorem for quasiconvex functions and we provide several examples of extremal quasiconvex functions, proving in particular a conjecture made by Šverák. We then further investigate, through numerical experiments, the relationship between rank-one convexity and quasiconvexity, particularly in low dimensions. We also give a concise proof of Ornstein's L1 non-inequality in low dimensions. This part of the thesis contains joint work with Daniel Faraco (Universidad Autónoma de Madrid) and Rita Teixeira da Costa (University of Cambridge). The third part of this thesis deals with low regularity problems for nonlinear underdetermined PDEs. We mostly focus on the prescribed Jacobian equation, although applications to energy-dissipative solutions of the incompressible Euler equations are also discussed. Concerning the prescribed Jacobian equation, we prove an ill-posedness result for the Dirichlet problem. We also study the uniqueness and symmetry properties of energy minimisers with prescribed Jacobian, concluding that in general they are non-unique and non-symmetric. These results answer several questions posed by Hélein, Hogan- Li-McIntosh-Zhang and Ye in the 1990s and provide some of the first results concerning low regularity solutions of the Jacobian equation. We also prove a nonlinear version of the classical Open Mapping Theorem from Functional Analysis. Our result applies to a wide range of PDEs and, in particular, it applies to the weakly continuous nonlinearities characterized in the first part of this thesis, of which the Jacobian determinant is a particular example. As consequences of this nonlinear Open Mapping Theo- rem, we prove: i) a partial selection criterion for solutions of the Jacobian equation in the critical Sobolev space, and ii) generic non-existence of weak solutions to the incompressible Euler equations over Rⁿ with fastly decaying kinetic energy. This part of the thesis contains joint work with Lukas Koch (University of Oxford) and Sauli Lindberg (Aalto University).

Computational homogenization is adopted to assess the homogenized two-dimensional response of periodic composite materials where the typical microstructural dimension is not negligible with respect to the structural sizes. A micropolar homogenization is, therefore, considered coupling a Cosserat medium at the macro-level with a Cauchy medium at the micro-level, where a repetitive Unit Cell (UC) is selected. A third order polynomial map is used to apply deformation modes on the repetitive UC consistent with the macro-level strain components. Hence, the perturbation displacement field arising in the heterogeneous medium is characterized. Thus, a newly defined micromechanical approach, based on the decomposition of the perturbation fields in terms of functions which depend on the macroscopic strain components, is adopted. Then, to estimate the effective micropolar constitutive response, the well known identification procedure based on the Hill-Mandel macro-homogeneity condition is exploited. Numerical examples for a specific composite with cubic symmetry are shown. The influence of the selection of the UC is analyzed and some critical issues are outlined.

We give a comprehensive presentation of the periodic unfolding method for perforated domains, both when the unit hole is a compact subset of the open unit cell and when this is impossible to achieve. In order to apply the method to boundary-value problems with nonhomogeneous Neumann conditions on the boundaries of the holes, the properties of the boundary unfolding operator are also extensively studied. The paper concludes with applications to such problems and examples of reiterated unfolding.

When considering an effective, i.e. homogenized description of waves in periodic media that transcends the usual quasi-static approximation, there are generally two schools of thought: (i) the two-scale approach that is prevalent in mathematics and (ii) the Willis’ homogenization framework that has been gaining popularity in engineering and physical sciences. Notwithstanding a mounting body of literature on the two competing paradigms, a clear understanding of their relationship is still lacking. In this study, we deploy an effective impedance of the scalar wave equation as a lens for comparison and establish a low-frequency, long-wavelength dispersive expansion of the Willis’ effective model, including terms up to the second order. Despite the intuitive expectation that such obtained effective impedance coincides with its two-scale counterpart, we find that the two descriptions differ by a modulation factor which is, up to the second order, expressible as a polynomial in frequency and wavenumber. We track down this inconsistency to the fact that the two-scale expansion is commonly restricted to the free-wave solutions and thus fails to account for the body source term which, as it turns out, must also be homogenized—by the reciprocal of the featured modulation factor. In the analysis, we also (i) reformulate for generality the Willis’ effective description in terms of the eigenfunction approach, and (ii) obtain the corresponding modulation factor for dipole body sources, which may be relevant to some recent efforts to manipulate waves in metamaterials.

Our planet is facing significant changes of biodiversity across spatial scales. Although the negative effects of local biodiversity (α diversity) loss on ecosystem stability are well documented, the consequences of biodiversity changes at larger spatial scales, in particular biotic homogenization, that is, reduced species turnover across space (β diversity), remain poorly known. Using data from 39 grassland biodiversity experiments, we examine the effects of β diversity on the stability of simulated landscapes while controlling for potentially confounding biotic and abiotic factors. Our results show that higher β diversity generates more asynchronous dynamics among local communities and thereby contributes to the stability of ecosystem productivity at larger spatial scales. We further quantify the relative contributions of α and β diversity to ecosystem stability and find a relatively stronger effect of α diversity, possibly due to the limited spatial scale of our experiments. The stabilizing effects of both α and β diversity lead to a positive diversity–stability relationship at the landscape scale. Our findings demonstrate the destabilizing effect of biotic homogenization and suggest that biodiversity should be conserved at multiple spatial scales to maintain the stability of ecosystem functions and services.

This manuscript presents a novel approach allowing damped ultrasonic wave propagation analysis of textile composite structures modelled at a mesoscopic level (i.e. modelling the yarns and matrix distinctively). Current modelling approaches rely on material homogenisation for analysis at a macroscopic scale and thus overlook the effect of textile architecture on wave propagation. This work aims at predicting wave propagation characteristics in damped textile composite structures and the induced complex phenomena for applications in structural health monitoring. The developed methodology involves mesoscale modelling of a textile composite structure period using a specialised textile modeller for pre-processing as well as conventional finite element methods. This is combined with the periodic structure theory as well as a mode-based reduction method named Craig-Bampton allowing for solving a reduced eigenproblem deriving from the equation of motion. A multiscale approach is used throughout the thesis to enable the comparison of standard wave propagation analysis of composite structures, using homogenised properties, with the more complex analysis proposed in this thesis. The need for this methodology is demonstrated as well as its validity. The first axis of this thesis describes the methodology for undamped wave propagation analysis in textile composites. Its advantages, such as the prediction of complex phenomena and the possible applications, are thoroughly described and issues discussed. Its increased accuracy over macroscale prediction methods is exposed. A second axis of the thesis is experimental validation of the methodology by means of linear scans of waves measured by a laser vibrometer and generated by a piezoelectric transducer in 3D woven composite samples. It is shown that the numerical mesoscale methodology provides accurate predictions. The third axis is the prediction of dispersion characteristics in large layered assemblies of textile composites. An attempt toward homogenisation of textile composites using a dispersion curves inversion technique based on genetic algorithms is proposed for this purpose. It is concluded that complex textile composites cannot be approximated by simple macroscale models. The last axis of the thesis introduces a damping model to predict the frequency dependent loss factor of waves propagating in these textile composite structures. The strong influence of mesoscale architecture over loss factor is demonstrated.

Human activities have reorganized the earth's biota resulting in spatially disparate locales becoming more or less similar in species composition over time through the processes of biotic homogenization and biotic differentiation, respectively. Despite mounting evidence suggesting that this process may be widespread in both aquatic and terrestrial systems, past studies have predominantly focused on single taxonomic groups at a single spatial scale. Furthermore, change in pairwise similarity is itself dependent on two distinct processes, spatial turnover in species composition and changes in gradients of species richness. Most past research has failed to disentangle the effect of these two mechanisms on homogenization patterns. Here, we use recent statistical advances and collate a global database of homogenization studies (20 studies, 50 datasets) to provide the first global investigation of the homogenization process across major faunal and floral groups and elucidate the relative role of changes in species richness and turnover. We found evidence of homogenization (change in similarity ranging from −0.02 to 0.09) across nearly all taxonomic groups, spatial extent and grain sizes. Partitioning of change in pairwise similarity shows that overall change in community similarity is driven by changes in species richness. Our results show that biotic homogenization is truly a global phenomenon and put into question many of the ecological mechanisms invoked in previous studies to explain patterns of homogenization.

Collisional models are a category of microscopic framework designed to study open quantum systems. The framework involves a system sequentially interacting with a bath comprised of identically prepared units. In this regard, quantum homogenization is a process where the system state approaches the identically prepared state of bath unit in the asymptotic limit. Here, we study the homogenization process for a single qubit in the non-Markovian collisional model framework generated via additional bath-bath interaction. With partial swap operation as both system-bath and bath-bath unitary, we numerically demonstrate that homogenization is achieved irrespective of the initial states of the system or bath units. This is reminiscent of the Markovian scenario, where partial swap is the unique operation for a universal quantum homogenizer. On the other hand, we observe that the rate of homogenization is slower than its Markovian counter part. Interestingly, a different choice of bath-bath unitary speeds up the homogenization process but loses the universality, being dependent on the initial states of the bath units.