Variational eigenvalue approximation of non-coercive operators with application to mixed formulations in elasticity

We present an abstract framework for the eigenvalue approximation of a class of non-coercive operators. We provide sufficient conditions to guarantee the spectral correctness of the Galerkin scheme and to obtain optimal rates of convergence. The theory is applied to the convergence analysis of mixed finite element approximations of the elasticity and Stokes eigensystems.

Beyond Turing: far-from-equilibrium patterns and mechano-chemical feedback

Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator–inhibitor interaction destabilized by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternatives to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-à-vis the classical Turing analysis, and give an overview of recent results and future challenges for both approaches. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Non-well-ordered lower and upper solutions for semilinear systems of PDEs

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

MAPPING AND ENHANCING DESIGN STUDIES WITH PSI META-THEORETIC DESIGN FRAMEWORK

Design research as a field has been studied from diverse perspectives starting from product inception to their disposal. The product of these studies includes knowledge, tools, methods, processes, frameworks, approaches, and theories. The contexts of these studies are innumerable. The unit of these studies varies from individuals to organizations, using a variety of theoretical tools and methods that have fragmented the field, making it difficult to understand the map of this corpus of knowledge across this diversity.In this paper, we propose a model-based approach that on the one hand, does not delve into the details of the design object itself, but on the other hand, unifies the description of design problem at another abstraction level. The use of this abstract framework allows for describing and comparing underlying models of published design studies using the same language to place them in the right context in which design takes place and to enable to inter-relate them, to understand the wholes and the parts of design studies.Patterns of successful studies could be generated and used by researchers to improve the design of new studies, understand the outcome of existing studies, and plan follow-up studies.

Labeled Bipolar Argumentation Frameworks

An essential part of argumentation-based reasoning is to identify arguments in favor and against a statement or query, select the acceptable ones, and then determine whether or not the original statement should be accepted. We present here an abstract framework that considers two independent forms of argument interaction—support and conflict—and is able to represent distinctive information associated with these arguments. This information can enable additional actions such as: (i) a more in-depth analysis of the relations between the arguments; (ii) a representation of the user’s posture to help in focusing the argumentative process, optimizing the values of attributes associated with certain arguments; and (iii) an enhancement of the semantics taking advantage of the availability of richer information about argument acceptability. Thus, the classical semantic definitions are enhanced by analyzing a set of postulates they satisfy. Finally, a polynomial-time algorithm to perform the labeling process is introduced, in which the argument interactions are considered.

A theory of spectral partitions of metric graphs

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

Beyond Turing: Far-from-equilibrium patterns and mechano-chemical feedback

Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator-inhibitor interaction destabilised by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternative to the ‘classical’ Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-à-vis the ‘classical’ Turing analysis, and give an overview of recent results and future challenges for both approaches.

Ultra-lightweight Authentication

In this chapter we provide a critical look at the state of the art in ultra-lightweight authentication protocols. We start by outlining the features of the current ubiquitous and pervasive computing environment that have motivated the development of the ultra-lightweight paradigm which uses only basic arithmetic and logical operations. We emphasize its goals and its main challenges. Then, we focus our attention on the authentication problem. We use an abstract framework for modeling the protocols proposed over the years, in order to discuss their design strategies and the security and privacy properties they aim to achieve. After that, we survey the weaknesses and the common pitfalls in both the design and the analysis of ultra-lightweight authentication protocols. Finally, we conclude the chapter by discussing some fundamental ideas and research directions.