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This chapter introduces the useful model-analytic tools of both sensitivity analysis and structured singular value analysis and their application to cellular networks. It reviews the Nyquist stability criterion and extends it to conditions guaranteeing robust stability (RS). It then examines the structured singular value analysis for robust performance. This chapter shows that control-theoretic tools, such as sensitivity analysis and phase sensitivity, offer powerful means for network elucidation and manipulation in biological systems. It suggests that using tools from control theory to guide both mathematical modeling and experimental design can facilitate the iterative paradigm of systems biology and shed light on the complex network behavior underlying biological organisms.

This chapter discusses crystal structure analysis, which is usually based on diffraction phenomena caused by the interaction of matter with X-rays, electrons, or neutrons. Although the theory of diffraction is substantially the same for all types of radiation, X-ray scattering is considered with particular interest. Diffraction phenomena can be conveniently described if one is able to model the radiation, model the crystal, and model the interaction between radiation and crystal. The last item is the focus of this chapter: the description of the simplest interaction — scattering by an electron — is followed in sequence by scattering of atoms, molecules, unit cell, and crystal. An essential tool for the treatment is the knowledge of the properties of the Fourier transform.

The competition between chemical and spatial constraints leads to the unusual properties of many materials that have been analysed using the bond valence model. Applications are found in fields ranging from crystallography, i.e, structure analysis (Sections 12.2 and 12.8), through condensed matter physics (Section 12.3), chemistry (Section 12.4), mineralogy (Section 12.5), materials science (Section 12.6) and molecular biology (Section 12.7). They have been particularly useful in the analysis of perovskite related solids (Section 12.3.1) and the modelling of the surfaces between inorganic solids and aqueous solutions (Section 12.6.2).

This chapter discusses structural equation modeling (SEM), also referred to as causal modeling and covariance structure analysis, which is used to evaluate the consistency of substantive theories with empirical data. SEM is a hybrid model that integrates path analysis and factor analysis. SEM is related to factor analysis because it may be used to test hypothesized relationships between unmeasured or latent variables and observed or empirical indicators of latent variables. SEM is related to path analysis because it may be used to test hypothesized relationships between constructs. Thinking of SEM as a combination of factor analysis and path analysis ensures consideration of SEM's two primary components: the measurement model and the structural model.

In this chapter, the two main classes of intermetallic QCs known so far are introduced: decagonal QCs and icosahedral QCs. The terminology “decagonal” and “icosahedral”, respectively, refers to the Laue symmetry (10/m, 10/mmm and m3¯5¯, respectively) of their diffraction patterns (intensity weighted reciprocal lattice) or, equivalently, to the symmetry of the interatomic vector map (auto-correlation function or Patterson map). It also refers to the “bond-orientational order” of a QC structure, what is nothing else but its vector map. The full spacegroup symmetry of a quasiperiodic structure can best be described in the framework of the nD approach. However, an equivalent description is also possible in 3D reciprocal space based on the symmetry relationships between the complex structure factors.

To understand the structural ordering of aperiodic crystals such as incommensurately modulated phases, composite crystals, quasicrystals and their structurally closely related rational approximants, the higher-dimensional (nD) approach is essential. In a properly selected nD space, aperiodic crystal structures can be described as 3D sections/projections of nD periodic “hypercrystal” structures. Furthermore, “non-crystallographic” symmetries of quasicrystals can become compatible with nD lattice symmetry. For instance, 5-, 8-, 10-, and 12-fold rotational symmetries are proper symmetry operations in 5D and icosahedral point group symmetries in 6D hypercubic lattices. First the so-called “strip-projection” (or “cut-and-project”) method is introduced, because it allows a more intuitive understanding of when and how clusters order quasiperiodically. Then the nD section method is presented, which is the method of choice for nD structure analysis, because it can make use of the reciprocal space information experimentally accessible by diffraction methods.

As conventional cadastral maps only show building perimeters, they contain no information about a city’s internal structure—that is, about the complex interplay of architecture and its social and economic engagement. Urban planning seems to have little consideration for what goes on inside the buildings lining a street. The method of Urban Parterre Modeling described in this chapter refers to the city’s ground floor as a holistic urban system, covering both built-up and non-built-up areas. Street, ground floor, and inner courtyard are treated as separate entities, which brings out the interrelations between them, as it becomes clear that actual and potential ground-floor uses directly impact the correlated public street space. The chapter describes the development of the model and its present and future applications.