bluuebunny/crossref_metadata_2025 huggingface dataset:
Introduction
Authors: Forster T E
Published: November 2023

It is often assumed that Russell’s paradox shows that the universe cannot be a set, but this follows only if every subclass of a set is a set. So, if we want the universe to be a set (hereafter “V E V”), we must somehow forestall the application of comprehension-like schemes to big2 sets, lest we enable the diagonal construction to go through. This does not exclude the possibility that, for some suitable notion of “small”, the small sets in such a system form a model of ZF. In some of the systems described here this possibility is made explicit in an axiom whose relative consistency can be proved. Thus the extra generality of V E V need not cost anything.
Permutation Models
Authors: Forster T E
Published: November 2023

The first thing to say about this permutation method is that it is not the permutation method of Fraenkel and Mostowski. It is due to Rieger [1957] and Bernays [1954] and was first applied to NF by Scott [1962]. It is familiar to students of ZF as the technique used in the standard proof of the independence of the axiom of foundation.
Interaction Of Two Black Holes In The Slow-Motion Limit
Authors: D'eath P D
Published: November 2023

Following the ideas of Chapter 3, in this chapter we shall describe a method of analysing the interaction of two black holes in a wide variety of physical situations (D’Eath 1975b). The black holes may be rotating and are permitted in our approach to have comparable masses, subject to two types of restriction on the interaction. First, it is assumed that the spatial separation of the black holes is large compared to the radii of their event horizons, so that each black hole is in the far field of the other. Second, we suppose that the relative velocity of the black holes is small compared to the speed of light, so that their combined gravitational field can be described in terms of a slow-motion approximation scheme. Then we should expect that the dominant interaction between them is just that between a pair of Newtonian point particles; it is further assumed in a slow-motion scheme that the Newtonian potential energy of the interaction is comparable to the relative kinetic energy. This allows one to consider a bound system of two black holes moving at non-relativistic speeds. It also allows one to study a quasiNewtonian scattering of two black holes at non-relativistic speeds in which the scattering angle may be quite large (Ruffini and Wheeler 1971; Hansen 1972). These two types of interaction are described by the same equations. The aim is then to find more and more accurate approximations to the dynamical spacetime geometry, subject to the vacuum Einstein field equations, and from this description of the geometry to find more and more accurate approximation to the equations of motion and spin propagation of the black holes.
Axisymmetric Black-Hole Collisions At The Speed Of Light: Perturbation Treatment Of Gravitational Radiation
Authors: D'eath P D
Published: November 2023

As remarked in Chapter 5, in the many years since general relativity was originally formulated by Einstein, no-one has found, owing to the complexity and nonlinearity of the field equations, any physically realistic analytic solution which does not possess a large number of simplifying symmetries. However, if one is interested in studying the generation of gravitational radiation by realistic physical sources, then one must of necessity consider isolated gravitating systems that are time-dependent and which can have no simplifying features apart from axisymmetry. General reviews of the subject of gravitational waves are given by Thorne (1977, 1987).
Axisymmetric Black-Hole Collisions At The Speed Of Light: Gravitational Radiation - Results And Conclusions
Authors: D'eath P D
Published: November 2023

In Chapters 6 and 7, the axisymmetric collision of two black holes was studied in the case in which the black holes approach at the speed of light, each with energyµ in the centre-of-mass frame. A perturbation approach was used, in which a large Lorentz boost is applied to the incoming data, so that the energies. λ, ν of the resulting incoming shock waves obey λ « ν. The subsequent evolution is described by a singular perturbation problem, with small parameter (.λ/ ν). When one boosts back to the centre-of-mass frame, successive terms of the perturbation series provide information on gravitational radiation which propagates at small angles from the symmetry axis near the curved shock 2 which has been distorted and deflected in the collision. (By symmetry, the same holds for radiation near the backward direction.) The gravitational radiation is described by the news function (Bondi et al. 1962) which is expected to have the convergent expansion (eqn (6.1.3))
On the Classification of Homotopy Types
Authors: Baues Hans-Joachim
Published: November 2023

In this chapter we describe fundamental new results on the classification of homotopy types. On the one hand, we get a classification by Postnikov invariants (k-invariants); on the other hand, we obtain a classification by boundary invariants. The general properties of these invariants lead us to introduce algebraic concepts which we call ‘kype functors’ E and ‘bype functors’ F, respectively. Here kype is a new word derived from the words k-invariant and type and similarly bype is derived from boundary invariant and type.
Linear algebra
Authors: Meise Reinhold , Vogt Dietmar
Published: November 2023

Since the intersection of linear subspaces of a vector space E is again a linear subspace, there exists, corresponding to each subset M of E, a smallest linear subspace, span(M), which contains M. This subspace span(M) is called the linear hull of M.
Bidual and reflexivity
Authors: Meise Reinhold , Vogt Dietmar
Published: November 2023

If E is a normed space, then not only its dual space E′ but especially also the dual space (E′′; of E′; are of interest. We refer to (E′′; =: E″ as the bidual of E. We shall now consider E″.
Self-adjoint extensions
Authors: Meise Reinhold , Vogt Dietmar
Published: November 2023

The main results of the earlier chapter can be applied to those symmetric operators which are restrictions of self-adjoint operators. Therefore it is of interest to know which symmetric operators admit a self-adjoint extension. We shall now deal with this question. In this context let H be a complex Hilbert space.
Locally convex vector spaces
Authors: Meise Reinhold , Vogt Dietmar
Published: November 2023

In this chapter we introduce concepts in locally convex vector spaces which are fundamental for the entire part and concern ourselves with their elementary properties.
Duality theory of locally convex spaces
Authors: Meise Reinhold , Vogt Dietmar
Published: November 2023

As we have already seen, in general there exists no distinguished topology on the dual space E′; of a locally convex space E, except in the case of normed spaces. We shall therefore explore in this section possible and meaningful locally convex topologies on E′;. Thereby we shall also establish that certain notions in E do not depend on the topology given on E, rather only on the dual space E′;.
A splitting theorem
Authors: Meise Reinhold , Vogt Dietmar
Published: November 2023

We require some preparation for the proof of the splitting theorem. We begin with the introduction of certain scales of Hilbert spaces.
Subspaces and quotients of s
Authors: Meise Reinhold , Vogt Dietmar
Published: November 2023

In this chapter we shall apply the splitting theorem of chapter 30 to characterize the closed subspaces, the quotients and the complemented subspaces of s. More-over, we shall show that many Fréchet spaces of C∞-functions are isomorphic to s.
Auxiliary Facts from Mathematical Elasticity
Authors: Kozlov Vladimir , Maz’ya Vladimir , Movchan Alexander
Published: November 2023

In this section we present definitions and notations from linear elasticity. For more details we refer to the books by Westergaard (1952), Lurie (1970), Atkin and Fox (1980) and Gurtin (1981).
Non-Degenerate Elastic Multi-Structures
Authors: Kozlov Vladimir , Maz’ya Vladimir , Movchan Alexander
Published: November 2023

The analysis of the elasticity problems in Chapter 4 shows that for multi-structures consisting of a set of parallel thin rods and a three-dimensional finite body the resistance to loads applied along thin rods is much larger than the resistance to forces associated with bending of thin rods. That is, with applied loads of the same order, the transverse displacements of the multi-structure are much larger compared with the order displacements along the rods. This phenomenon is connected with the special geometry of the multi-structure we dealt with.
Enumeration
Authors: Read Ronald C
Published: November 2023

In this chapter we present the basic problem of graphical enumeration-the enumeration of unlabelled graphs. In particular, we look at Pólya‘s Enumeration Theorem and some problems to which it can be applied. We then consider the enumeration of some other kinds of graphs. The chapter concludes with some remarks about unsolved problems of graphical enumeration. How many graphs are there on four vertices? This question can easily be resolved by a process of trial and error, but if we ask how many graphs there are on ten vertices, the answer can be reliably found only by theoretical means, while the similar question for an indefinite number of vertices clearly requires a formula of some kind. In this chapter we shall be largely concerned with deriving such a formula, and some similar results.
Groups
Authors: Cameron Peter J
Published: November 2023

In this chapter, the connections between a graph and its automorphism group are described. The main themes are that most graphs have very little symmetry; abstract automorphism groups can be prescribed independently of most graph- theoretic properties; vertex-transitivity, however, entails various structural properties; and still higher degrees of symmetry can be expected to lead to a complete classification. The most important connection between graphs and groups is the fact that every graph G has an automorphism group, consisting of all permutations of the vertex set which map edges to edges and non-edges to non-edges. For most graphs, the automorphism group is trivial, consisting of only the identity permutation. However, every group is the automorphism group of some graph, and highly symmetric graphs have a number of special properties not shared by arbitrary graphs.
Probability
Authors: Mcdiarmid Colin
Published: November 2023

What are ‘typical’ properties of a graph? How can random methods yield existence results for graphs? What will randomized graph algorithms do? What have isoperimetric inequalities for graphs to do with probability? This chapter gives an introductory taste of the fertile area of graphs and probability. Since Paul Erdős brought together the subjects of graph theory and probability some forty years ago, the relationship has flowered marvellously.
Genetic markers for population studies-II Molecular markers
Authors: Burnett John
Published: November 2023

A wide range of natural molecules exist such as particular secondary metabolites like the lichen acids which are produced by the fungal component, the whole protein content or specific polymorphic enzyme proteins. Unfortunately, characteristic species-specific molecules only occur sporadically amongst the fungi so that their use as markers is limited, although they are often characteristic of higher groups such as genera, sections or families.
The generation of variation-I Mutation and migration
Authors: Burnett John
Published: November 2023

Strictly, mutation is the sole source of new variation, although it is recombination (see Chapter 7) which mainly generates novel multilocus genotypes. Mutations can occur in both nDNA and mtDNA, and the latter are quite common in fungi. Before the structure of DNA had been elucidated, mutations were recognized by the phenotypic changes they caused.
Fungal species
Authors: Burnett John
Published: November 2023

Mycology as a recognizable activity only really got under way in the eighteenth century and then as a branch of botany rather than as a science in its own right. Therefore fungal taxonomists inevitably adopted the procedures and concepts developed by botanical taxonomists and did their best to fit the fungi into a framework largely developed for flowering plants.
Student years and Greenland
Authors: Kruuk Hans
Published: November 2023

Leiden was only some 15 km from The Hague, a short trip by train, and Niko studied from home. He did not have to leave his stamping grounds and his friends, and for him the change from school to university was not the upheaval that it was for most other students. But what he found himself studying was something different, and totally new territory.
The classification of surfaces
Authors: Lawson Terry
Published: November 2023

In this chapter we discuss surfaces and classify them up to homeomorphism. A surface is a topological space which locally looks like a piece of the plane, such as a sphere or the exterior of a donut (a torus). Before specializing to surfaces, we begin by introducing the concept of an n-manifold, where a surface is a 2-manifold. Manifolds constitute one of the primary areas of study in topology. Our study of surfaces will introduce us to many of the key ideas in manifold theory in a fairly concrete geometric setting.
Sketches of an Elephant A Topos Theory Compendium
Authors: Johnstone Peter T
Published: November 2023

Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and thereby to demonstrate the overall unity of the subject. The material is organized in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text.
D3
Authors: Johnstone Peter T
Published: November 2023

We saw in Section D1.4 that, for each cartesian (resp. regular, coherent, geometric) theory 𝕋, there is a category C𝕋 of the appropriate kind containing a model of 𝕋 which is generic, in the sense defined there. Since we are mainly interested in models which live in toposes, it would be convenient to be able to construct a topos containing a model of a given theory which is generic amongst models in toposes. There are various methods for doing this. One, which was sketched in Section B4.2, relies on the 2-categorical structure of (Grothendieck) toposes, in order to build up (the topos containing) the generic model by an inductive process. Here we shall give a more ‘pedestrian’, but also more explicit, method, which has the advantage that the explicit description of the generic model which we obtain allows us to establish results about models of a theory by first proving them in the generic case, and then transferring them along inverse image functors.
The experimental side
Authors: Balucani Umberto , Zoppi Marco
Published: November 2023

In Chapter 1 we have provided some intuitive arguments for restricting our attention to a limited class of liquids, namely those approximately described by the Hamiltonian (1.1), (1.2). Such a picture of ‘simple’ liquids deserves a more detailed discussion if we wish to make contact with the real liquids which can be investigated experimentally. In principle, a liquid can be described as a state of matter occurring in a limited range of pressures and densities, where it exhibits peculiar macroscopic properties (absence of a definite shape, very low compressibility, fluidity, etc.) well known from everyday experience. Microscopically, we deal with a system of interacting particles (henceforth referred to as the ‘molecules’). In a first idealization, we may assume that the molecules are individually characterized by a well- defined set of properties, independent of the overall state of the system. This assumption has important consequences even from an experimental point of view, since it implies that collective properties can be studied by assuming that the relevant ‘single-molecule’ parameters (such as the dipole moment or the polarizability) can be used as a basic set, independent of the thermodynamic conditions such as temperature and pressure.
Collective properties
Authors: Balucani Umberto , Zoppi Marco
Published: November 2023

In Section 3.4 we saw that all the results of ordinary Navier-Stokes hydrodynamics can formally be deduced by a microscopic memory function approach. In perspective, this success stems from the slow character of the hydrodynamic variables in the limit of small wave vectors, which in turn implies the validity of a Markovian ansatz for the corresponding memory functions. As a by-product of the approach, it was moreover possible to establish ‘Green-Kubo relations’ for the macroscopic transport coefficients, which can be expressed as time integrals of ordinary correlation functions.
The ideal gas temperature
Authors: Carrington Gerald
Published: November 2023

We shall introduce first the properties of an ideal gas, starting with the ideal gas law and the elementary kinetic theory of gases. In effect we begin by taking a microscopic approach because most beginners feel comfortable with the basic concepts involved. Remember that the primary purpose of this chapter, and the two which follow, is not to develop the physical properties of ideal gases as such. Instead we shall use the ideal gas as an example to introduce some important thermodynamic ideas. An ideal gas is one in which the long-range interactions between the atoms or molecules can be ignored. I assume that you know the atoms of the gas move about incessantly, colliding with the walls of the containing vessel and with each other. In an ideal gas the duration of these encounters is short compared with the time the atoms spend in free motion between collisions.
Zeroth and first laws
Authors: Carrington Gerald
Published: November 2023

In Chapters 2-4 we introduced a number of thermodynamic functions for ideal gases. Because we knew the primary equations of state, such as (2.1.1) and (2.2.2), it was not necessary to introduce the underlying laws of thermodynamics in a formal way. Instead we explored some basic thermodynamic concepts without the need to establish the general theory first. In this chapter we shall develop the notions of states and processes further. The aim is to enlarge upon ideas that are already familiar. Although we might proceed in a number of ways, we shall follow a traditional route which is based on the classical laws of thermodynamics. This approach acknowledges how firmly the classical laws are embedded in the literature and language of physics.
Superconducting states
Authors: Kuramoto Y , Kitaoka Y
Published: November 2023

Superconductivity, one of the best understood many-body problems in physics 11 J, again became a challenging problem when a new kind of superconducting phenomenon was discovered in CeCu2Siz by Steglich et al. [2] in 1979. The system behaves as a heavy-electron material close to magnetic instability. In the subsequent decade, intensive investigations of a class of uranium compounds established a new field of heavyelectron superconductivity by successive discoveries of superconductivity in UBe13, UPl3, URu2Si2, UPd2Al3 and UNi2Al3 [3,4]. The most important characteristics for a series of uranium heavy-electron superconductors, except for UBen, are that superconductivity coexists with antiferromagnetism and that the specific heat coefficient γ lies in a broad range from 700 mJ/(mol K2) (lJBe13) to 60 mJ/(mol K2) (URu2Si2). The f-shell electrons, which are strongly correlated by Coulomb repulsive interaction, determine the properties of heavy quasi-particles at the Fermi level.
Elastic Scattering
Authors: Boffi Sigfrido , Franco Davide Pacati Carlotta Giusti , Radici Marco
Published: November 2023

Measuring the angular distribution of an elastically scattered probe provides the first information on a system and, in particular, on its size, charge and/or matter distribution, magnetization density, and the role of its constituents. In the case of electron scattering this information is contained in the differential cross section in terns of the so-called electromagneticforn factors. In photon scattering electric polarizability and magnetic susceptibility of the target are the relevant quantities which represent its dynamic response. This chapter collects the formalism for elastic electron scattering which is the basis for further development and comparison in the following chapters dedicated to various inelastic processes. The role and meaning of the electromagnetic form factors come out very naturally in this context and are discussed in a number of cases, with particular emphasis on the nucleon form factors. This is because nucleon form factors play an important role when investigating the response of atomic nuclei. Without a good knowledge of the behaviour of free nucleons it would be dangerous to draw any conclusion about the nuclear response.
Inclusive Inelastic Scattering
Authors: Boffi Sigfrido , Franco Davide Pacati Carlotta Giusti , Radici Marco
Published: November 2023

Inelastic electron scattering has proved to be a very precise tool for studying the excitation spectrum of atomic nuclei owing to the possibility of simultaneously varying energy and momentum transfer. Since the very beginning of experimentation a large amount of work has been done in determining the energies and properties of excited states. Single-particle and collective excitations were identified and helped to understand the structure and dynamics of nuclei in terms of the motion of the constituent nucleons. We refer to the literature cited in Chapter 1 for a comprehensive account of the subject and to Fig. 1.1 for a schematic view of the nuclear response in the energy-momentum plane. Here we are interested in the nuclear response over a range of energies well above the region of giant resonances and below that of deep inelastic scattering, where two distinct features appear as large peaks with characteristic dependence on energy and momentum transfer.
Semi-Inclusive Inelastic Scattering
Authors: Boffi Sigfrido , Franco Davide Pacati Carlotta Giusti , Radici Marco
Published: November 2023

Semi-inclusive inelastic reactions are produced when one or more particles are ejected from the target and are detected in coincidence with the scattered electron, while the residual nucleus remains undetected. If the dynamical variables of all the final particles are determined, we have an exclusive reaction. The simplest case of semi-inclusive reaction is one-nucleon knockout. The kinematic variables of this process and the notation are shown in Fig. 6.1. In this situation, when the residual nucleus is left in its ground state or in a discrete excited state, we have an exclusive reaction.
Perspectives
Authors: Boffi Sigfrido , Franco Davide Pacati Carlotta Giusti , Radici Marco
Published: November 2023

Electron scattering and photon absorption at intermediate energies have been shown to provide rich information on nucleon interaction and nuclear structure. With presentday experimentation a large domain of the (ω, q)-plane is accessible to study the hadron response under a variety of kinematical conditions. Taking advantage of the unique precision of the electromagnetic probe this response can be studied in terms of measurable form factors and structure functions. Spin degrees of freedom can also be exploited, increasing the number of measured observables and enabling us to access the transition amplitudes of the process and therefore to obtain a complete determination of the scattering matrix. The continuous progress of experimental research both in detection systems and in the beam characteristics, the increasing precision of the measurements and the number of experimental facilities under construction will make available in the near future a large collection of new and appealing data.
Structure of molecules
Authors: Pettifor D G
Published: November 2023

Molecules show a wide variety of structure, the number of variants increasing dramatically with number of atoms in the molecule. Thus, whereas the dimer displays only one structure type, the trimer can be either linear like CO2 or bent like H20. Molecules with four atoms, on the other hand, can take the one-dimensional linear chain structures like H4, the two-dimensional square ring structure like S4 or the rhombic close-packed planar structure like Si4, the three-dimensional tetrahedral structure like P4, or numerous other distorted structural variants. The five-atom molecule, as we have already seen in Fig. 1.15, chooses the close-packed plane as its ground state structure for Na5, the trigonal bipyramid for Mg5 and Si5, the regular pyramid for Al5, and the envelope structure for S5 respectively. The six-atom molecule can choose between the one-dimensional linear chain, the two-dimensional hexagonal ring or close-packed plane, the three-dimensional pentagonal pyramid, trigonal tripyramid, trigonal prism, or octahedron-to name but a few! The energy difference between these competing structure types is usually very small, often being of the order of 1% of the binding energy or less. Hence, the reliable prediction of the ground-state structure of a particular molecule requires self-consistent quantum mechanical calculations of high precision that expend large amounts of computer time.
Bonding of sp-valent metals
Authors: Pettifor D G
Published: November 2023

A metal is often pictured as a gas of free electrons into which has been immersed a lattice of positive ions. The metallic bond is, therefore, thought of as having no directional character in contrast to saturated covalent bonds with their hybrid orbitals resistant to bond bending. The metal atoms behave like hard spheres that are held together by the all-pervasive electron glue, taking close-packed structures such as fee, bee, or hep. Unlike valence compounds with their restrictive requirement of electron-pair bonding between neighbouring sites, metals can form alloys over a wide range of composition, atoms of one type replacing those of another with comparative ease within the electron gas. We will see in this chapter that this conventional view of the metallic bond is indeed an excellent description of sp-valent metals such as sodium, magnesium, and aluminium. We will begin, therefore, by linking the world of small molecules and extended solids together by applying the jellium model to a study of cohesion in atomic clusters. We will find that it predicts special stability for alkali metal clusters containing magic numbers of atoms that correspond to electronic shell closings. However, as we have already seen in Chapter 2, jellium is only in equilibrium at one specific electron density. The underlying ionic lattice is required for differentiating between the elements and predicting the properties of the sp-valent metals. The influence of a periodic crystalline potential on the electronic structure is introduced through the one-dimensional Kronig-Penney model that illustrates all the essential features of band theory. The key concept of the pseudopotential is presented in order to account for the well-known but surprising fact that the free-electron gas is only very weakly perturbed by the ionic lattice. This allows us to develop a quantitative nearly free electron (NFE) model of the metallic bond that is entirely consistent with the usual picture of a metal as a gas of free electrons into which has been immersed a lattice of positive ions.
Structure of sp-valent metals
Authors: Pettifor D G
Published: November 2023

The description of the metallic bond in the previous chapter assumed that the free-electron gas remained homogeneously distributed as the pseudopotential lattice of ion cores was immersed within it. In practice, of course, the free electrons will respond to screen the ion cores, thereby leading to a modulation of the electron density. This modulated electron density then acts in turn on the underlying ionic lattice to give rise to a structuredependent contribution in the total energy. We will see in this chapter that this small structurally dependent contribution to the binding energy can be evaluated quantitatively using second-order perturbation theory within the nearly free electron (NFE) approximation.
Diffuse Neutron Scattering From Crystalline Materials
Authors: Nield Victoria M , Keen David A
Published: November 2023

The present book is an accessible, comprehensive guide to diffuse neutron scattering, an important technique for studying structural disorder in materials. The text takes the reader through theoretical, computational and experimental developments in the subject and describes in detail its application to a number of structural disorder problems. These include the more traditional subjects of substitutional disorder in alloys and orientational disorder in molecular systems as well as the more recent studies of superionic and framework materials. Particular emphasis is placed on recent refinement methods for data interpretation which are compared with established computer simulation techniques and analytical approaches. The book collects disparate themes into one unique volume, written as an introduction to the method for graduate scientists. It will be a valuable reference text for any crystallographer keen to understand and apply modern interpretative techniques to diffuse scattering data.
Introduction
Authors: Nield Victoria M , Keen David A
Published: November 2023

Diffuse scattering occurs from all crystalline materials at all temperatures. It arises from any temporary or permanent breakdown of space group symmetry, and so contains information on the deviations from the average structure. Information on all the different types of disorder in crystalline solids is present, and interpretation of the diffuse scattering is necessary for a detailed understanding of local structural information.
Computer simulation and modelling
Authors: Nield Victoria M , Keen David A
Published: November 2023

It is becoming increasingly common for scientists to adopt computer simulation methods to investigate crystalline systems. This is principally for two reasons, both of which are essentially pragmatic. First, scientists are more likely to be investigating far more complex materials, including those materials which display disorder. It is often not possible to apply established theoretical descriptions to such systems analytically and, as has been shown in Chapter 3, it very quickly becomes a complex task to describe the interactions between atoms or spins rigorously as the material becomes increasingly disordered. Even model systems which can be described very simply may be difficult to solve exactly. Computer simulations have sometimes been termed ‘computer experiments’ and in some senses they are just that: the computer contains a well defined small system which is constrained to obey specific interactions and from which physical properties of the system are ‘measured’ using thermodynamic principles or other appropriate averaging. However computer simulations have another role since they can also be used to bridge the gap between what can be expressed theoretically and what can be measured experimentally. They can be used to check the validity of approximations made when developing theoretical expressions by applying them to systems of real interest. They also allow the scientist to ‘see’ the effect of analytical parameters in terms of ‘real’ atom positions and correlations. This is especially true in the case of structural disorder where a more complete picture may be obtained from a computer simulation than from an analytical treatment relying on a number of possibly inappropriate approximations.
Alloys and other binary materials
Authors: Nield Victoria M , Keen David A
Published: November 2023

In the previous chapters we have built up our understanding of the experimental and theoretical methods applicable to alloys and binary systems. In the present chapter we review a wide selection of studies on these materials. These are largely neutron scattering studies, taken from the recent literature. However, where appropriate, X-ray studies which supplement the neutron work, or are in some sense novel or ground breaking, have been included. Space has not allowed a significant amount of detail to be included on the background to the studies, and the interested reader is referred to the original papers for further insight.
Green-Function Methods
Authors: Davison Sydney G , Stęślicka Maria
Published: November 2023

Of all the methods discussed so far, none is as mathematically sophisticated and elegant as the Green-function (GF) ones to be described here in their various forms. Because these techniques are both powerful and versatile, it is important to master them, despite their more abstract formalism, which tends to make them somewhat more difficult to understand. The great advantage of GF techniques is that they provide direct access to the density of states (DOS) to which many physical properties are related. For example, the intensity of photoemission from solids is proportional to the DOS. Indeed, most spectroscopic yields are linked to the initial DOS. Other properties, such as bond order (Phillips 1969), can also be related to the GF.
Elastic Scattering of Ions by Atoms
Authors: Bransden B H , McDowell M R C
Published: November 2023

In this chapter a short account of the theory of ion atom elastic scattering will be given on the assumption that the interaction between two atoms (or ions) A and B can be represented by a real central energy-independent potential W(R), where R = RA R8 is the position vector of A relative to B. This model can be applied to describe the relative motion of A and B approximately even if the collision is inelastic (capture, excitation, or ionization), provided the impact energy is large compared with changes in the internal electronic energies of the atoms. This is usually the case when the impact energy is of the order of 100 eV or greater.
Charge Exchange at Very Low Energies the Perturbed Stationary State Model
Authors: Bransden B H , McDowell M R C
Published: November 2023

In this chapter we shall discuss charge exchange at energies for which the relative velocity of the colliding ions is very small compared with typical orbital velocities of the pseudomolecule composed of the active electron moving in the field of the ionic cores A and 8. A natural choice of basis functions in a coupled channel model for such a collision is a set of adiabatic molecular orbitals, modified by suitable translation factors as in equations (3.99). The coupled equations, determining the channel wave functions using the coordinate set (s, R) where R is the vector joining A to 8 and s is the position vector of the electron relative to C, the centre of mass of A and B, are then of the form (3.93), the overlap and potential matrix elements being defined by the expressions (3.91) and (3.94). In the perturbed stationary state model, which in the past was employed for the majority of calculations at very low energies, the translation factors which allow for the change in translational motion of the electron on capture are neglected completely. This model, originally developed by Massey and Smith (1933) is described in Section 4.1. At very low energies the number of significant partial waves is limited and a partial-wave expansion of the coupled equations is indicated.
Many-Electron Systems
Authors: Bransden B H , McDowell M R C
Published: November 2023

As we have already noted in Sections 4.7 and 5.9, the coupled channel methods developed in Chapter 3 for quasi-one-electron systems can be generalized, in principle, for systems with any number of electrons. The basis functions can be adiabatic molecular wave functions which diagonalize the electronic Hamiltonian, or atomic functions (and pseudostates) which diagonalize the unperturbed atomic Hamiltonians in the different arrangements. In the former case all the interactions including the electronelectron interaction are taken into account and in the latter all the internal interactions in the separate atoms are allowed for in the basis. Calculating such basis functions for a many-electron system with sufficient accuracy is already a demanding task; nevertheless there have been a growing number of investigations of two-electron or quasi-two-electron systems, some of which have been illustrated in Chapter 6. To some extent, the difficulties of constructing many-particle molecular wave functions can be overcome by using a basis in which electron electron correlation is absent. Correlation can then be taken into account when forming the coupled equations. This approach, which is due to Hare] and Salin, will be discussed below. Apart from the calculation of basis functions, the computational effort required to treat two-electron systems can be very great, compared with one-electron systems, for several reasons. For example, the elements of the interaction matrix become more difficult to compute. In particular, with an atomic basis composed of travelling Slater orbitals the computation of the matrix elements of the electron-electron interaction are particularly time consuming. If specifically two-electron processes are important, for example double capture, double excitation or ionization, simultaneous capture and excitation, or simultaneous capture and ionization, the size of the basis required to represent the two-electron wave function is much greater than that required for one-electron systems.
Solutions to exercises
Authors: Maggiore Michele
Published: November 2023

Since photons are massless the only energy scale is provided by kBT.
Pseudo-equilibrium replica analysis
Authors: Coolen A C C
Published: November 2023

The first serious attempt in literature at solving an MG model using statistical mechanical techniques dealt with the simplest case of having fake and inconsistent market histories, i.e. with ζ = 1 in equation (2.2) and with equation (2.4) describing fake history statistics, two strategies per agent (i.e. S = 2), and additive decision noise (2.12). It employed a strategy that had in the past proved effective in the study of non-equilibrium neural network models. The idea is to find a Lyapunov function H(q) for the MG process, with q = (q  1  ,..., qN), which would allow one to determine the stationary state properties via an analysis of the minima of H(q).
The overall bid distribution
Authors: Coolen A C C
Published: November 2023

At this stage we make a small detour. Rather than introducing and analyzing new MG versions, we pause briefiy and return to the intriguing simulation data on the overall bid statistics as shown for instance in Figs 1.2 (for the on-line MG with true market history) and 1.11 (for the online MG with ‘fake’ market history).
MG versions with new types of phase transitions
Authors: Coolen A C C
Published: November 2023

The generating functional method allows us for N →∞ to reduce the problem of solving the original coupled and disordered N -agent process to that of solving a stochastic and non-Markovian ‘effective single-agent’ equation, without disorder. So far our strategy in analyzing this single-agent equation has been to first look for ergodic states, described by TTI stationary solutions, with finte susceptibility χ. We assumed that, if such a solution exists (which was found to be true in each of the MG versions studied so far, for sufficiently large α), given time, the system will forget its initial conditions and approach this particular stationary state. Numerical simulations have so far confirmed this assumed scenario. The predictions for the persistent observables of the MG, as well as for the location of the point where this particular solution should break down (marked by a divergence of χ) and the resulting phase diagrams, agreed perfectly with our experimental observations.
Theory of stationary random processes
Authors: Diggle Peter J
Published: November 2023

We begin this chapter by recalling some notation and definitions from Chapters 1 and 2. A random function is a set of random variables {Y(t)}, in which the time-index, t can assume any real value. The trend of {Y(t)} is the non-random function µ(t) = E[Y(t)].
Forecasting
Authors: Diggle Peter J
Published: November 2023

Before discussing specific forecasting methods, we make the point that forecasting involves extrapolation in the presence of random variation, and as such is bound to be difficult in practice. The difficulties are threefold. Firstly, we need to be confident that the mechanisms generating the historical data will continue to apply in the future. Secondly, even if we can assume a stable generating mechanism, we need to express this mechanism in the form of a statistical model for the data. The difficulty here is that different models may fit the historical data equally well, yet give quite different predictions of the future. Thirdly, even supposing we have identified the correct statistical model, the statistical uncertainty inherent in any estimates of the model parameters will be magnified as predictions are made progressively further into the future.
Contingency tables
Authors: Lauritzen Steffen L
Published: November 2023

Before we begin developing the formal theory, it seems appropriate to consider a few examples that illustrate some basic points.
Multivariate normal models
Authors: Lauritzen Steffen L
Published: November 2023

The graphical models in this chapter have a particularly simple interpretation and a rather detailed statistical theory. The models assume that the variables observed follow a regular multivariate normal distribution. Conditional independence restrictions in the multivariate normal distribution can be expressed in a simple fashion through zero restrictions on the inverse covariance matrix.
Further topics
Authors: Lauritzen Steffen L
Published: November 2023

In this chapter we give a brief discussion of a number of topics that have not been covered in the central part of the book but nevertheless illuminate essential aspects of the theory and application of graphical models.
Simplified and two-stage quasi-likelihood estimators
Authors: Hutton J E , Ogunyemi, O T , Nelson Paul I
Published: November 2023

Properties of a quasi-likelihood estimator are typically obtained from an expansion of the quasi-likelihood in which the remainder term is asymptotically negligible. The remainder term, however, can be very difficult to control in some important applications, especially in models where both the mean and variance depend on the unknown parameter. In such cases we propose a modified quasi-likelihood based on the use of a simplified covariance matrix. Conditions for the consistency and asymptotic normality of the resulting estimators are developed. We illustrate this approach with a branching process with immigration model and present a simulation study of the small sample behaviour of the estimators.
Making use of a regression model for inferences about a finite population mean
Authors: Mantel H
Published: November 2023

Godambe and Thompson (1986) define and develop simultaneous optimal estimation of super-population and finite-population parameters based on a super-population model and a survey-sampling design. However, we may be interested in estimating a finite population mean that has no direct relationship to the finite-population parameter of Godambc and Thompson’s theory. In this paper we extend the super-population model in such a way that there is a direct relationship between the finite-population mean and the finite¬ population parameter. The resulting estimator of the finite-population mean is compared to the generalized regression estimator.
The glim language
Authors: Swan Tony , Gilchrist Robert , Bradley Malcolm , Clarke Mike , Green Peter , Reese Allan , Hinde John , Stalewski Andrew , O’brien Carl
Published: November 2023

This chapter outlines the basic components of the GLIM language and introduces the general syntax and mode of use of the package. GLIM statements are composed of characters from the GLIM character set, which is listed in the Reference Guide [13.1]. Essentially, this consists of upper- and lower-case letters, digits, and a range of operators and brackets which will be introduced in the relevant chapters of the User Guide. Lower-case letters are treated as equivalent to the corresponding upper-case letters in statements when they are interpreted by the GLIM program, except in strings when the case is retained. The character ‘@’ is used in fault messages to indicate an invalid input character.
File handling and utilities
Authors: Swan Tony , Gilchrist Robert , Bradley Malcolm , Clarke Mike , Green Peter , Reese Allan , Hinde John , Stalewski Andrew , O’brien Carl
Published: November 2023

GLIM has facilities for inputting statements and data from external files and similarly output can be directed to a file to provide hard copy output. These are accessed via secondary input and secondary output channels. Certain channels are automatically available to the user. One of these is for the transcript file which, by default, is a copy of everything that appears on the screen during an interactive session, including both input and output. Another is for the journal file which is a record of the GLIM statements input by the user during a session. This can be ‘replayed’ to reproduce a GLIM session. The user has control over what should be automatically stored in these files (see [4.3.1]). A third file which is available for input by the user is the macro library. This contains tried and tested macros which provide a valuable extension to the facilities available in GLIM [Appendix A]. There are also facilities for the user to save (to a dump file) the GLIM session in its current state and subsequently restart it at a later time, thus allowing an analysis to be spread over more than one session.
The Theory of Generalized Linear Models
Authors: Swan Tony , Gilchrist Robert , Bradley Malcolm , Clarke Mike , Green Peter , Reese Allan , Hinde John , Stalewski Andrew , O’brien Carl
Published: November 2023

In Chapter IO it was observed that the standard (univariate) statistical models can be thought of as being special cases of a more general structure, called a generalized linear model (GLM). This chapter introduces the underlying theory behind a GLIM analysis by defining more rigorously what is meant by a generalized linear model (GLM) and by outlining the algorithmic basis of the model fitting procedures of GLIM.
The PASS Facility
Authors: Swan Tony , Gilchrist Robert , Bradley Malcolm , Clarke Mike , Green Peter , Reese Allan , Hinde John , Stalewski Andrew , O’brien Carl
Published: November 2023

Execution of a PASS statement causes the keyword or integer, the values of the list of vector(s)/scalar(s), and the elements of the macro to be passed to the FORTRAN subroutine PASS. As distributed in the standard release of GLIM, the body of subroutine PASS includes some routines for illustrative purposes. It is anticipated, though, that the user may wish to supply their own body for PASS, which will perform certain operations on the input parameters that are not available (or not easily available) via the standard GLIM directives. This note explains how this may be achieved.
Inverse Natural Exponential Families on Jr
Authors: Seshadri V
Published: November 2023

Recall that in Chapter 1, Section 1.4 we promised an explanation of the term ‘inverse’ appearing in the inverse Gaussian distribution. We shall now offer an explanation and justification of this usage by introducing the concept of inverse pairs of distributions (measures) and natural exponential families on R The ideas were defined in Letac (1986b) who coined the term ‘reciprocity’ in true Gallic style (note that the French usage of the terms ‘reciproque’ and ‘inverse’ corrspond exactly to the terms ‘inverse’ and ‘reciprocal’ just another instance of inversion), in an attempt to provide significant examples of inverse variates first introduced in his terse one-page paper by Tweedie (1945).