bluuebunny/crossref_metadata_2025 huggingface dataset:
A realizability interpretation of Martin-Löf’s type Theory
Authors: Coquand Catarina
Published: November 2020

The notion of computability predicate developed by Tail gives a powerful method for proving normalization for various λ-calculi. There are actually two versions of this method: a typed and an untyped approach. In the former approach we define a computability predicate over well-typed terms. This technique was developed by Tail (1967) in order to prove normalization for Gödel’s system T. His method was extended by Girard (1971) for his System F and by Martin-Löf (1971) for his type theory. The second approach is similar to Kleene’s realizability interpretation (Kleene 1945), but in this approach formulas are realized by (not necessarily well-typed) λ-terms rather than Gödel numbers. This technique was developed by Tail (1975) where he uses this method to obtain a simplified proof of normalization for Girard’s system F. The method was later rediscovered independently by several others, for instance Mitchell (1986). There are two ways of defining typed λ-calculus: we have either equality as conversion (on raw terms) or equality as judgments. It is more difficult to show in the theory with equality as judgments that every well-typed term has a normal form of the same type. We can find different approaches to how to solve this problem in Altenkirch (1993), Coquand (1991), Goguen (1994) and Martin-Löf (1975). There are several papers addressing normalization for calculi with equality as untyped conversion; two relevant papers showing normalization are Altenkirch’s (1994) proof for system F using the untyped method and Werner’s (1994) proof for the calculus of constructions with inductive types using the other method. Another difference is the predicative and impredicative formulations of λ-calculi where it is more intricate to prove normalization for the impredicative ones. The aim of this paper is to give a simple argument of normalization for a theory with “real” dependent types, i.e. a theory in which we can define types by recursion on the natural numbers (like Nn). For this we choose to study the simplest theory that has such types, namely the N, Π, U-fragment of Martin-Löf’s polymorphic type theory (Martin-Löf 1972) which contains the natural numbers, dependent functions and one universe. This theory has equality as conversion and is predicative.
Variable-free formalization of the Curry-Howard Theory
Authors: Tait William W.
Published: November 2020

The reduction of the lambda calculus to the theory of combinators in Schönfinkel 1924 applies to positive implicational logic, i.e. to the typed lambda calculus, where the types are built up from atomic types by means of the operation A → B, to show that the lambda operator can be eliminated in favour of combinators К and S of each type A → (B → A) and(A → (B →C)) →((A →C)), respectively. I will extend that result to the case in which the types are built up by means of the general function type ∀х : A.B(x) as well as the disjoint union type Ǝх : A.B(x)— essentially to the theory of Howard 1980. To extend the treatment of→ to ∀ we shall need a generalized form of the combinators К and S, and to deal with Ǝ we will need to introduce a new form of the combinator S (whose type turns out to be a general form of the axiom of choice). But also in the present context, if we are to eliminate variables, then not only the lambda operator for forming terms, but also quantification as a variable-binding operation for forming formulas must be analyzed away; so we will need an analogue of the combinators for formulas. As usual, we shall write if for the value s(t) of the function s for the argument t; so rst is (r(s)))(t), etc. Let v be a free variable of type A. We wish to rewrite the formulas B(v), ∀x: A.B(x) and Ǝх : A.B(x), respectively, as B′v, ∀В′ and ƎB′, where B' is a type-valued function on A.
Nearly as nature intended: An exhibition of some curious molecules in the foods we eat
Authors: Emsley John
Published: November 2020

There are scores of myths surrounding the things we eat: chocolate is almost addictive; Coca-Cola is just a concoction of chemicals; garlic wards off heart disease and cancer; an aspirin a day keeps the doctor away. None of these statements is true, but they contain a germ of truth. In this gallery we can inspect the portraits of some of the natural and unnatural chemicals which a normal diet contains. The pleasures of eating are sweet but fleeting, while the warnings about food seem bitter and never-ending. The warnings we should heed are those of professional dietitians, the front-line troops who are fighting the war against poor nutrition and unbalanced diets. While they help the people who are referred to them, the rest of us only hear their advice second-hand, and even then we do not heed it—which may explain why one person in five is now classed as obese (33% or more overweight) i. the USA, and one in ten in Britain. Behind the front-line dietitians is a regiment of armchair food commanders who offer their advice to anyone who listens. Often it is soundly based, telling us how to lose weight and still be properly nourished, but a lot is rather unhelpful, merely condemning some popular foods as ‘junk’ without explaining why they are so (although this term is generally taken to mean that they contain too much sugar, salt, saturated fats and additives). Examples of junk food are chocolate, colas, hamburgers and french fries. Sadly the healthy alternatives, such as raw celery, mineral water and lentils, lack appeal for many, and especially for children. Alongside claims about junk food come more dire warnings about the chemicals that are present in other foods, and especially if these have been added merely to make food look and taste more tempting, or if they are there as contaminants that come from pesticides and processing. Surprisingly, most food-related illness comes not from these, but from micro-organisms such as bacteria and fungi, and we are most at risk when we eat food that has not been properly stored or prepared. Ideally food should be free of all dangerous impurities, be they bacteria, fungi or chemicals.
The origin of protocells
Authors: Maynard Smith John , Szathmary Eors
Published: November 2020

Cellularization has the following main aspects that we have to explain: • The need for active (self-generated) compartmentation when metabolism is liberated from the surface. • The origin of membranogenic molecules and membranes. • The origin and mechanism of spontaneous protocell fission. • The transportation problem. Simple membranes are not ‘leaky’ enough to permit important nutrients to pass through. • Were the first protocells autotrophs or heterotrophs? The evolution of the first autocatalytic metabolic cycle. • The iron-sulphur world and the RNA world: are they mutually exclusive or complementary? • The problem of the origin of the two membranes of negibacteria, the most ancient existing group of organisms. • The origin of chromosomes and DNA synthesis. We shall discuss these problems in turn. As we discussed before, the prebiotic pizza has the ability to localize metabolites and genes. This is advantageous for two reasons: Reactants are kept in each other’s proximity, which ensures that reaction rates will be high enough and that important compounds do not drift away. Genes will interact, directly (e.g. by influencing each other’s replication) or indirectly (by catalysing steps of metabolism), only with their neighbours: selection will thus be able to ensure cooperation among genes that would otherwise compete against each other. Life liberated itself from surfaces a long time ago. Somehow, passive localization must have been replaced by an active process of membrane generation, maintenance and fission. The basic structure of contemporary biomembranes is as follows. There is a molecular bilayer of lipids, to which proteins are attached in various ways. The bilayer is formed because the membrane constituents are so-called amphipathic molecules: they have a hydrophilic head and a hydrophobic tail. Since the binding interaction of water with itself is much stronger than that between water and hydrophobic compounds, the latter are expelled by water as much as possible; this results in tails coming together. A simple sheet of bilayer would still be not at the energy minimum because its edges would be exposed to water. An energetically favourable solution is the formation of a lipid vesicle.
The origin of eukaryotes
Authors: Maynard Smith John , Szathmary Eors
Published: November 2020

The basic structures of a bacterial and a eukaryotic cell are shown in Fig. 8.1. The differences whose origins call for an explanation are as follows: • The bacterial cell has a rigid outer cell wall, usually made of the peptidoglycan, murein. In eukaryotes, the rigid cell wall is not universal, and cell shape is maintained primarily by an internal cytoskeleton of filaments and microtubules. • Eukaryotic cells have a complex system of internal membranes, including the nuclear envelope, endoplasmic reticulum and lysosomes. • Bacteria have a single circular chromosome, attached to the rigid outer cell wall. In eukaryotes, linear chromosomes are contained within a nuclear envelope, which separates transcription from translation: communication between nucleus and cytoplasm is via pores in the nuclear envelope. • Eukaryotes have a complex cytoskeleton. The actomyosin system powers cell division, phagocytosis, amoeboid motion and the overall contractility to resist osmotic swelling. Microtubules and the associated motor proteins (kinesin, dynein and dynamin) ensure the accurate segregation of chromosomes in mitosis, ciliary motility and the movement of transport vesicles. Intermediate filaments form the structural basis for the association of the endomembranes and nuclear-pore complexes with the chromatin to form the nuclear envelope, while other intermediate filaments help to anchor the nucleus in the cytoplasm. One crucial difference between prokaryotes and most eukaryotes has been omitted from Fig. 8.1: this is the presence of mitochondria, and, in plants and algae, of chloroplasts. The reason for the omission is that, on the scenario for eukaryote origins that seems to us most plausible, these intracellular organelles originated later in time than the structures shown in the figure. The differences between these cell types justifies the recognition of two empires of life (above the kingdom level): Bacteria and Eukaryota (Cavalier-Smith, 199la; Table 8.1). (It is interesting that this taxonomic rank was recognized by Linnaeus.) Within each of the empires, there are two major categories: Bacteria consist of the kingdoms Eubacteria and Archaebacteria, and Eukaryota are divided into the superkingdoms Archaezoa and Metakaryota. The justification for these divisions is as follows. The Archaebacteria, in contrast to the Eubacteria, never have murein cell walls, and their single cell membrane contains isoprenoidal ether rather than acyl ester lipids.
The development of spatial patterns
Authors: Maynard Smith John , Szathmary Eors
Published: November 2020

It is convenient to start with an account of the development of the flower in the crucifer Arabidopsis (Coen & Meyerowitz, 1991), for several reasons. The development of plants is in one respect simpler than that of animals: cells do not move relative to one another. Much of animal development is achieved by cell movement, contraction and adhesion; these are processes we need not bother with when thinking about plants. As yet, less is known of Arabidopsis than of Drosophila or of vertebrates: what is known is simple and elegant, and brings into sharp focus the things that are not yet known. The flower develops from a disc of cells. Four concentric rings differentiate. The outermost ring gives rise to four sepals, the next innermost ring to four petals, the next to six stamens, and the central cells to two fused carpels. Mutants are known that alter this pattern. Usually, a given mutant alters two neighbouring rings of organs. A simple model explains these mutants. Genes are expressed in three regions, A, B and C. If we symbolize the corresponding genes by a, b and c, then a gives rise to sepals, ab to petals, be to stamens and c to carpels. This would account for normal development, but if we are also to explain the mutants, we must assume that genes a and c are mutually inhibitory, so that when a is absent or deficient, c is expressed over the whole region, and vice versa. This model has been confirmed in two ways. First, it can satisfactorily explain double mutants. Triple mutants, in which all three gene activities are missing, develop leaf-like structures in all four rings: this confirms the longheld view that flower organs are modified leaves. The phenotypes of the various double mutants can be predicted from the model, and are as expected. The model has been confirmed more directly by hybridization in situ of the RNA transcripts of the different genes, which appear early in the development of the flower primordium in the expected places. An almost identical system is present in Antirrhinum. There is molecular homology between the genes in the two species.
Development and evolution
Authors: Maynard Smith John , Szathmary Eors
Published: November 2020

In the nineteenth century, ideas about development, heredity and evolution were inextricably mixed up, because it seemed natural to suppose that changes that first occurred in development could become hereditary, and so could contribute to evolution. This was not only Lamarck’s view but Darwin’s, expressed in his theory of pangenesis. Weismann liberated us from this confusion, by arguing that information could pass from germ line to soma, but not from soma to germ line. If he was right, geneticists and evolutionary biologists could treat development as a black box: transmission genetics and evolution could be understood without first having to understand development. Since Weismann, developmental biology has had only a rather marginal impact on evolutionary biology. One day, we have promised ourselves, we will open the box, but for the time being we can get along very nicely without doing so. Recent progress in developmental genetics, some of which has been reviewed in the last three chapters, oblige us to reopen the question. In fact, there are three related questions, not one. The first, which is most relevant to the theme of this book, is the ‘levels of selection’ question: why does not selection between the cells of an organism disrupt integration at the level of the organism? This is the topic of section 15.2. The second is the problem of the inheritance of acquired characters. This old problem has reappeared in a new guise. We now recognize the existence of cell heredity, mediated by different mechanisms from those concerned with transmitting information between generations. In section 15.3, we discuss whether cell heredity plays any role in evolutionary change. Finally, in sections 15.4 and 15.5, we ask whether recent molecular information sheds any light on another old problem—that of the extraordinary conservatism of morphological form, maintained despite dramatic changes of function. This conservatism has led anatomists to identify a small number of basic archetypes, or bauplans. There is little doubt that conservatism is real. Consider, for example, the fact that bones and cartilages, which in humans serve in swallowing, sound production and hearing, are derived from elements of the gill apparatus whereby our fish ancestors exchanged gases with seawater, and, before that, in all probability, from elements of a filter-feeding apparatus.
Computational Anatomy: Hypothesis Testing on Disease
Authors: Grenander Ulf , Miller Michael I.
Published: November 2020

Computational Anatomy (CA) is the study of human anatomy I ∈ I = Iα ◦G, an orbit under groups of diffeomorphisms of anatomical exemplars Iα ∈ I. The observable images I<D ∈ ID are the output of Medical imaging devices. This chapter focuses on the third of the three components of CA: (iii) generation of probability laws of anatomical variation P(·) on the images I within the anatomical orbits and inference of disease models. The basic random objects studied are the diffeomorphic transformations encoding the anatomical objects in the orbit; Gaussian random fields are constructed based on empirical observations of the transformations. Hypothesis testing on various neuromorphometric changes are studied.
Basic Theory
Authors: Gautschi Walter
Published: November 2020

This introductory chapter is to present a quick review of material on orthogonal polynomials that is particularly relevant to computation. Proofs of most results are included; for those requiring more extensive analytic treatments, references are made to the literature.
Living and fossil placentals
Authors: Kemp T.S.
Published: November 2020

The vast majority of living and fossil mammals are placentals. Today there are about 4,400 species, which are traditionally organised into 18 Orders, with an extra one if the Pinnipedia are separated from the Carnivora, and a twentieth if the recently extinct Malagasy order Bibymalagasia is recognised as such. There have been many attempts to discover supraordinal groupings from amongst these Orders based on morphological characters, though few proposals have been universally accepted. It is only with the advent of increasingly large sets of molecular sequence data in the last few years that a reasonably robust resolution looks imminent, although these contemporary analyses are remarkably and controversially at odds with the traditional ones. Novacek et al. (1988) summarised the then current situation regarding supraordinal classification of placentals, a time at which morphology was still dominant but molecular data was at the threshold of significance. They accepted a basal group Edentata that combined the Xenarthra of the New World with the Pholidota of the Old, based on a few cranial characters, loss of the anterior teeth, and reduction of the enamel of the remaining ones. This left the rest of the living placentals as a monophyletic group Epitheria, sharing such apparently minor characters as the shape of the stapes bone in the ear. They found very little resolution within the Epitheria, and concluded that there was a polychotomy of no less than nine lineages arranged as a ‘star’ phylogeny. No remnant of the previously recognised taxon Ferungulata, created by Simpson (1945) for the Carnivora plus the ungulate orders Artiodactyla, Perissodactyla, Proboscidea, Hyracoidea, Sirenia, and Tubulidentata remained. On the other hand, three supra ordinal taxa of earlier authors did survive. One was Gregory’s (1910) Archonta, consisting of generally conservative forms and by now composed of the Primates, Dermoptera, Scandentia, and Chiroptera, but excluding the Lipotyphla. The second was Glires, originating with Linnaeus (1758) and widely accepted ever since, for the Rodentia and Lagomorpha; Novacek et al. (1988) tentatively placed the Macroscelidea as the sister-group of the Glires. The third supraordinal taxon recognised was, like Glires, well-established if not universally accepted.
Cosmology and the Arrow of Time
Authors: Penrose Roger , Gardner Martin
Published: November 2020

Central to our feelings of awareness is the sensation of the progression of time. We seem to be moving ever forward, from a definite past into an uncertain future. The past is over, we feel, and there is nothing to be done with it. It is unchangeable, and in a certain sense, it is ‘out there’ still. Our present knowledge of it may come from our records, our memory traces, and from our deductions from them, but we do not tend to doubt the actuality of the past. The past was one thing and can (now) be only one thing. What has happened has happened, and there is now nothing whatever that we, nor anyone else can do about it! The future, on the other hand, seems yet undetermined. It could turn out to be one thing or it could turn out to be another. Perhaps this ‘choice’ is fixed completely by physical laws, or perhaps partly by our own decisions (or by God); but this ‘choice’ seems still there to be made. There appear to be merely potentialities for whatever the ‘reality’ of the future may actually resolve itself to be. As we consciously perceive time to pass, the most immediate part of that vast and seemingly undetermined future continuously becomes realized as actuality, and thus makes its entry into the fixed past. Sometimes we may have the feeling that we even have been personally ‘responsible’ for somewhat influencing that choice of particular potential future which in fact becomes realized, and made permanent in the actuality of the past. More often, we feel ourselves to be helpless spectators - perhaps thankfully relieved of responsibility - as, inexorably, the scope of the determined past edges its way into an uncertain future. Yet physics, as we know it, tells a different story. All the successful equations of physics are symmetrical in time. They can be used equally well in one direction in time as in the other. The future and the past seem physically to be on a completely equal footing. Newton’s laws, Hamilton’s equations, Maxwell’s equations, Einstein’s general relativity, Dirac’s equation, the Schrödinger equation - all remain effectively unaltered if we reverse the direction of time.
Transformations
Authors: Newnham Robert E.
Published: November 2020

Many physical properties depend on direction and the resulting anisotropy is best described with the use of tensors. Tensors are classified according to how they transform from one coordinate system to another. Therefore, we begin by describing transformations. There are several reasons why we want to do this: (1) transformations help us define tensors, (2) these tensors can be used to describe physical properties, (3) the effects of symmetry on physical properties can be determined by howthe tensor transforms under a symmetry operation, (4) the magnitude of a property in any arbitrary direction can be evaluated by transforming the tensor, (5) using these numbers, we can draw a geometric representation of the property, and (6) the transformation procedure provides a way of averaging the properties over direction. This is useful when relating the properties of polycrystalline materials to those of the single crystal. Mathematically, there is nothing fancy about these transformations. We are simply converting one set of orthogonal axes (Z1, Z2, Z3) into another (Z'1, Z'2, Z'3). The two sets of axes are related to one another by nine direction cosines: a11, a12, a13, a21, a22, a23, a31, a32, and a33. Collectively all nine can be written as aij where i, j = 1, 2, 3. The axes and direction cosines are illustrated in Fig. 2.1. It is important not to confuse the subscripts of the direction cosines. As defined in the drawing, a12 is the cosine of the angle between Z'1 and Z2, whereas a21 is the cosine of the angle between Z'2 and Z1. The first subscript always refers to the “new” or transformed axis. The second subscript is the “old” or original axis. The original or starting axes is usually a right-handed set, but it need not be. The transformed “new” axes may be either right- or left-handed, depending on the nature of the transformation. This will become clearer when we look at some transformations representing various symmetry operations. In any case, both the old and new axes are orthogonal.
Acoustic waves II
Authors: Newnham Robert E.
Published: November 2020

Acoustic impedance, acoustic losses, acoustic waves in piezoelectric solids, and surface waves are discussed in this chapter, along with a number of nonlinear acoustic phenomena. The reflection and transmission of acoustic waves across a boundary is governed by acoustic impedance. One of the most important boundary value problems in acoustics concerns a plane wave incident upon a planar surface, dividing one medium from another. In the general case of an anisotropic medium, the incident beam consists of three waves (one quasilongitudinal, two quasitransverse), each traveling at a different velocity. Each of the three incident waves will be refracted and reflected at the boundary. If the second medium is also anisotropic, each incident wave will generate three reflected waves and three refracted waves, a total of 27 waves in all. Wave propagation in a polycrystalline solid where there are many grain boundaries becomes very complicated. The simpler case of a pure longitudinally-polarized wave at normal incidence to the boundary provides insight into the more general problem. In this case the reflection and transmission coefficients are governed by the relatively simple acoustic impedance parameter (ρc)1/2 = ρv, where ρ is the density, c the stiffness coefficient, and v the phase velocity. The reflection coefficient R at the interface between medium I and medium II is The MKS unit for acoustic impedance is the Rayl (=kg/m2 s). Atypical value for a solid is about 107 rayls. In many acoustic applications it is desirable to reduce reflection by matching the acoustic impedance of the two media. Lithium tantalate transducers are well-matched to iron, for example. Sound transmission from the transducer to the medium can be enhanced with composite materials or with graded coupling layers. Backing materials are often selected to promote reflection. In this case acoustic impedances are mismatched. Tungsten and air are two commonly used backing materials. In an isotropic material the acoustic impedance is (ρc11)1/2 for longitudinal waves and (ρc44)1/2 for shear waves. For anisotropic materials the wave velocities and acoustic impedance change with direction as indicated earlier.
Photoelasticity and acousto-optics
Authors: Newnham Robert E.
Published: November 2020

The change in refractive indices with mechanical stress, the photoelastic effect, is used in analyzing stress patterns in engineering components of complicated shape. It is also important in acousto-optic devices, optical switches, modulators, and scanners in which ultrasonic waves modulate the refractive index, producing an optical grating. Materials with sizable photoelastic coefficients (p) are required to enhance the interaction between mechanical strain x and refractive index n. Changes in the indicatrix are given by These quantities are actually tensors but are treated as scalars in the following discussion which is concerned with the magnitudes of the photoelastic coefficients p, and not their variation with direction. Unlike the linear electro-optic effect, photoelasticity occurs in all symmetry classes and is not a null property. Photoelastic coefficients are dimensionless because strain and refractive index are dimensionless. For most oxides and halides, pmax ≅ 0.2. The maximum values of p measured for other materials range from 0.1 to 0.6. To gain a clearer understanding of the effects of stress on refractive index, consider the effect of hydrostatic pressure on a cubic crystal.
Historical background
Authors: Hallam Tony
Published: November 2020

Georges Cuvier has not been treated with much respect in the English-speaking world for his contributions to the study of Earth history. Charles Lyell is thought to have effectively demolished his claims of episodes of catastrophic change in the past, and it is only in the past few decades, with the rise of so-called ‘neocatastrophism’, that a renewed interest has emerged in his writings, which date from early in the nineteenth century. Cuvier was a man of considerable ability, who quickly rose to a dominant position in French science in the post-Napoleonic years. Though primarily a comparative anatomist, his pioneer research into fossil mammals led him into geology. He argued strongly for the extinction of fossil species, most notably mammoths, mastodons, and giant sloths, at a time when the very thought of extinctions was rather shocking to conventional Christian thought, and linked such extinctions with catastrophic changes in the environment. This view is expressed in what he called the ‘Preliminary Discourse’ to his great four-volume treatise entitled Recherches sur les Ossements Fossiles (Researches on fossil bones), published in 1812. This extended essay was immensely influential in intellectual circles of the western world, was reissued as a short book, and was repeatedly reprinted and translated into the main languages of the day. It became well known in the English-speaking world through the translation by the Edinburgh geologist Robert Jameson (1813), who so bored the young Charles Darwin with his lectures that he temporarily turned him off the subject of geology. According to Martin Rudwick, who has undertaken a new translation which is used here, Jameson’s translation is often misleading and in places downright bad. It was Jameson’s comments rather than Cuvier’s text that led to the widespread belief that Cuvier favoured a literalistic interpretation of Genesis and wished to bolster the historicity of the biblical story of the Flood. The English surveyor William Smith is rightly credited with his pioneering recognition of the value of fossils for correlating strata, which proved of immense importance when he produced one of the earliest reliable geological maps, of England and Wales, but the more learned and intellectually ambitious Cuvier was the first to appreciate fully the significance of fossils for unravelling Earth history.
The influence of humans
Authors: Hallam Tony
Published: November 2020

We saw in Chapters 5 and 7 that the Quaternary was a time of low extinction rates despite a succession of strong environmental changes induced ultimately by climate. This began to change from a few tens of thousands of years ago with the arrival on our planet of Homo sapiens sapiens, which can be translated from the Latin as the rather smug ‘ultrawise Man’. It is widely accepted today that the Earth is undergoing a loss of species on a scale that would certainly rank in geological terms as a catastrophe, and has indeed, been dubbed ‘the sixth mass extinction’. Although the disturbance to the biosphere being created in modern times is more or less entirely attributable to human activity, we must use the best information available from historical, archaeological, and geological records to attempt to determine just when it began. Towards the end of the last ice age, known in Europe as the Würm and in North America as the Wisconsin, the continents were much richer in large mammals than today: for example, there were mammoths, mastodonts, and giant ground sloths in the Americas; woolly mammoths, elephants, rhinos, giant deer, bison, and hippos in northern Eurasia; and giant marsupials in Australia. Outside Africa most genera of large mammals, defined as exceeding 44 kilograms adult weight, disappeared within the past 100,000 years, an increasing number becoming extinct towards the end of that period. This indicates that there was a significant extinction event near the end of the Pleistocene. This event was not simultaneous across the world, however: it took place later in the Americas than Australia, and Africa and Asia have suffered fewer extinctions than other continents. There are three reasons for citing humans as the main reason for the late Pleistocene extinctions. First, the extinctions follow the appearance of humans in various parts of the world. Very few of the megafaunal extinctions that took place in the late Pleistocene can definitely be shown to pre-date the arrival of humans. There has, on the other hand, been a sequence of extinctions following human dispersal, culminating most recently on oceanic islands. Second, it was generally only large mammals that became extinct.
Elements of quantum theory
Authors: Barnett Stephen
Published: November 2020

We have seen that there is an intimate relationship between probability and information. The values we assign to probabilities depend on the information available, and information is a function of probabilities. This connection makes it inevitable that information will be an important concept in any statistical theory, including thermodynamics and, of course, quantum physics. The probabilistic interpretation of quantum theory has probability amplitudes rather than probabilities as the fundamental quantities. This feature, together with the associated superposition principle, is responsible for intrinsically quantum phenomena and gives quantum information theory its distinctive flavour. We shall see that the quantum rules for dynamical evolution and measurement, together with the existence of entangled states, have important implications for quantum information. They also make it possible to perform tasks which are either impractical or impossible within the classical domain. In describing these we shall make extensive use of simple but fundamental ideas in quantum theory. This chapter introduces the mathematical description of quantum physics and the concepts which will be employed in our study of quantum information.
Thermodynamic theory of solutions
Authors: Bokstein Boris S. , Mendelev Mikhail I. , Srolovitz David J.
Published: November 2020

A solution is a homogeneous mixture consisting of two or more components in which the composition can be continuously varied (within some range) with no change of phase. Solutions can be gases, liquids, or solids. We have already considered the properties of gaseous solutions (when we considered a mixture of ideal gases). In this chapter, we focus on condensed phases (i.e. liquids and solids). The composition of a solution can be described in several ways. Here are the most common: 1. The molar fraction of the ith component, xi, is the ratio of the number of moles of component i, ni, to the total number of moles of all species within the solution, n: 2. The weight fraction of the ith component, [i], is the ratio of the mass of component i, wi, to the total mass of all species within the solution, w: the weight fraction is often written as a weight percent [wt%] = 100[i]. 3. The molarity of the ith component, ci, is the number of moles of component i, ni per liter of solution, V: 4. The molality of ith component (used only for dilute solutions), mi, is the number of moles of component i, in 1 kg of solvent. There are several other definitions used to describe the composition of a solution, but we shall only refer to those described above in this text. The reason that there are so many definitions of the composition is related to how the term ‘‘concentration’’ is applied. For example, from the physical–chemical point of view, the molar fraction is the most convenient definition of the concentration since it is on an atomic basis. However, from the point of view of someone who has to prepare solutions from separate solutes and solvents, the mass fraction is the most convenient definition since it is directly related to the mass of the components, rather than the number of moles of the component. The former is easily measured, while the latter requires the additional knowledge of the molecular weight (and a trivial calculation).
Phase equilibriaII
Authors: Bokstein Boris S. , Mendelev Mikhail I. , Srolovitz David J.
Published: November 2020

In Chapter 2, we considered phase equilibria in a single component system. In the previous chapter, we considered the thermodynamics of multicomponent solutions. We are now prepared to discuss phase equilibria in multi-component systems. We consider the case of phase equilibria in a two-component systems in which all phases are condensed (i.e. solids or liquids). Under normal laboratory conditions experience suggests that the free energy of condensed phases is only very weakly dependent on the pressure and, hence, pressure is usually assumed to be constant. At constant pressure, the Gibbs phase rule (Eq. 2.3) for a two-component system takes the following form: . . . F = 3 – p, (4.1) . . . where F is the number of degrees of freedom (i.e. the number of independent parameters that must be set in order to fully determine the state of a system) and P is the number of phases present in equilibrium. This equation implies that the maximum number of independent parameters is 2; corresponding to the single-phase case. The most commonly used parameters are the temperature and the composition of the alloy. F = 2 implies that these two parameters can be varied over a finite range while the system remains the same, single phase. If two phases are in equilibrium with each other (P = 2), F = 1. This implies that the same two phases will be in equilibrium as we vary one parameter (the other parameter is not free). More specifically, we can retain the same two phase equilibrium while varying the alloy concentration over some range, if the temperature is described by some function, T(x2). T(x2) describes the curve (locus of points) along which this two-phase equilibrium occurs. Therefore, it is convenient to represent the phase equilibria in two-component condensed systems in T-x2 coordinates (as done below). Equation (4.1) implies that the maximum number of phases which can be in equilibrium with each other is three. This corresponds to F = 0—any change of temperature and/or alloy composition will lead to the disappearance of at least one of the phases.
Propositional logic
Authors: Hedman Shawn
Published: November 2020

In propositional logic, atomic formulas are propositions. Any assertion will do. For example, . . . A = “Aristotle is dead,” B = “Barcelona is on the Seine,” and C = “Courtney Love is tall” . . . are atomic formulas. Atomic formulas are the building blocks used to construct sentences. In any logic, a sentence is regarded as a particular type of formula. In propositional logic, there is no distinction between these two terms. We use “formula” and “sentence” interchangeably. In propositional logic, as with all logics we study, each sentence is either true or false. A truth value of 1 or 0 is assigned to the sentence accordingly. In the above example, we may assign truth value 1 to formula A and truth value 0 to formula B. If we take proposition C literally, then its truth is debatable. Perhaps it would make more sense to allow truth values between 0 and 1. We could assign 0.75 to statement C if Miss Love is taller than 75% of American women. Fuzzy logic allows such truth values, but the classical logics we study do not. In fact, the content of the propositions is not relevant to propositional logic. Henceforth, atomic formulas are denoted only by the capital letters A, B, C,. . . (possibly with subscripts) without referring to what these propositions actually say. The veracity of these formulas does not concern us. Propositional logic is not the study of truth, but of the relationship between the truth of one statement and that of another. The language of propositional logic contains words for “not,” “and,” “or,” “implies,” and “if and only if.” These words are represented by symbols: . . . ￢ for “not,” ∧ for “and,” ∨ for “or,” → for “implies,” and ↔ for “if and only if.” . . . As is always the case when translating one language into another, this correspondence is not exact. Unlike their English counterparts, these symbols represent concepts that are precise and invariable. The meaning of an English word, on the other hand, always depends on the context.
Proof theory
Authors: Hedman Shawn
Published: November 2020

As with any logic, the semantics of first-order logic yield rules for deducing the truth of one sentence from that of another. In this chapter, we develop both formal proofs and resolution for first-order logic. As in propositional logic, each of these provides a systematic method for proving that one sentence is a consequence of another. Recall the Consequence problem for propositional logic. Given formulas F and G, the problemis to decide whether or not G is a consequence of F. From Chapter 1, we have three approaches to this problem: • We could compute the truth table for the formula F → G. If the truth values are all 1s then we conclude that F → G is a tautology and G is a consequence of F. Otherwise, G is not a consequence of F. • Using Tables 1.5 and 1.6, we could try to formally derive G from {F}. By the Completeness Theorem for propositional logic, G is a consequence of F if and only if {F} ├ G. • We could use resolution. By Theorem1.76, G is a consequence of F if and only if ∅ ∈ Res(H) where H is a formula in CNF equivalent to (F ∧￢G). Using these methods not only can we determine whether one formula is a consequence of another, but also we can determine whether a given formula is a tautology or a contradiction. A formula F is a tautology if and only if F is a consequence of (A∨￢A) if and only if ￢F is a contradiction. In this chapter, we consider the analogous problems for first-order logic. Given formulas φ and ψ, how can we determine whether ψ is a consequence of φ? Equivalently, how can we determine whether a given formula is a tautology or a contradiction? We present three methods for answering these questions. • In Section 3.1, we define a notion of formal proof for first-order logic by extending Table 1.5. • In Section 3.3, we “reduce” formulas of first-order logic to sets of formulas of propositional logic where we use resolution as defined in Chapter 1.
Semantics for Disjunctive and Normal Disjunctive Logic Programs
Authors: Lobo Jorge , Minker Jack
Published: November 2020

During the past 20 years, logic programming has grown from a new discipline to a mature field. Logic programming is a direct outgrowth of work that started in automated theorem proving. The first programs based on logic were developed by Colmerauer and his students [Colmerauer et al., 1973] at the University of Marseilles in 1972 where the logic programming language PROLOG was developed. Kowalski [1974] published the first paper that formally described logic as a programming language in 1974. Alain Colmerauer and Robert Kowalski are considered the founders of the field of logic programming, van Emden and Kowalski [van Emden and Kowalski, 1976] laid down the theoretical foundation for logic programming. In the past decade the field has witnessed rapid progress with the publication of several theoretical results which have provided a strong foundation for logic programming and extended the scope of logic as a programming language. The objective of this article is to outline theoretical results that have been developed in the field of logic programming with particular emphasis to disjunctive logic programming. Disjunctive logic programming is an extension of logic programming and is useful in representing and reasoning with indefinite information. A disjunctive logic program consists of a finite set of implicitly quantified universal clauses of the form: . . . A1 , . . . , Am ← B1 , . . . , Bn m > 0 and n ≥ 0 (1) . . . where the Ai’s and the Bj’S are atoms. The atoms in the left of the implication sign form a disjunction and is called the head of the formula and those on the right form a conjunction and is called the body of the formula. The formula is read as “A1 or A2 or ... or Am if B1 and B2 and ... and Bn.” There are several forms of the formula that one usually distinguishes. If the body of the formula is empty, and the head is not, the formula is referred to as a fact. If both are not empty the formula is referred to as a procedure.
Non-Stoichiometric Compounds Derived From Point Defects
Authors: Kosuge Koji
Published: November 2020

In this chapter, we discuss ‘classical’ non-stoichiometry derived from various kinds of point defects. To derive the phase rule, which is indispensable for the understanding of non-stoichiometry, the key points of thermodynamics are reviewed, and then the relationship between the phase rule, Gibbs’ free energy, and non-stoichiometry is discussed. The concentrations of point defects in thermal equilibrium for many types of defect structure are calculated by simple statistical thermodynamics. In Section 1.4 examples of non-stoichiometric compounds are shown referred to published papers. The technical term ‘non-stoichiometric compounds’ has been used for a long time, in contradiction to the term ‘stoichiometric compounds’. The existence of non-stoichiometric compounds, which have also been called Bertholides compounds, cannot be explained from the law of definite proportion in its simplest meaning. Proust insisted that only stoichiometric compounds (also named Daltonide compounds) existed, whereas Bertholet maintained the existence of not only stoichiometric compounds but also non-stoichiometric compounds. This is a very famous argument in the history of chemistry. In the early years of the twentieth century, Kurnakov investigated the physical and chemical properties of intermetallic compounds in detail and found that the maximum or minimum in melting point, electrical resistivity, and also in the ordering temperature of lattices does not necessarily appear at the stoichiometric composition. An important discovery of Dingman was that stoichiometric FeO1.00 is non-existent under ordinary conditions. (At present, we can synthesize stoichiometric FeO1.00 under high pressure.) Non-stoichiometry, which originates from various kinds of lattice defect, can be derived from the phase rule. As an introduction, let us consider a trial experiment to understand non-stoichiometry (this experiment is, in principle, analogous to the one described in Section 1.4.8). Figure 1.1 shows a reaction vessel equipped with a vacuum pump, pressure gauge for oxygen gas, pressure controller for oxygen gas, thermometer, and chemical balance. The temperature of the vessel is controlled by an outer-furnace and the vessel has a special window for in-situ X-ray diffraction. A quantity of metal powder is placed on the chemical balance, and then the vessel is evacuated at room temperature.
Introduction
Authors: Gray John S. , Elliott Michael
Published: November 2020

As the oceans cover 70% of the earth’s surface, marine sediments constitute the second largest habitat on earth, after the ocean water column, and yet we still know more about the dark side of the moon than about the biota of this vast habitat. The primary aim of this book is to give an overview of the biota of marine sediments from an ecological perspective—we will talk of the benthos, literally the plants and animals at the bottom of the sea, but we will also use the term to include those organisms living on the intertidal sediments, the sands and muds of the shore. Given that most of that area is below the zone where light penetrates, the photic zone, the area is dominated by the animals and so we will concentrate on this component. Many of the early studies of marine sediments were taxonomic, describing new species. One of the pioneers was Carl von Linnaeus (1707–1778), the great Swedish biologist who developed the Linnaean classification system for organisms that is still used today (but under threat from some molecular biologists who argue that the Linnaean system is outdated and propose a new system called Phylocode). Linnaeus described hundreds of marine species, many of which come from marine sediments. The British marine biologist Edward Forbes was a pioneer who invented the dredge to sample marine animals that lived below the tidemarks. Forbes showed that there were fewer species as the sampled depth increased and believed that the great pressures at depths meant that no animals would be found deeper than 600 m. This was disproved by Michael Sars who in 1869 used a dredge to sample the benthos at 600 m depth off the Lofoten islands in Norway. Sars found 335 species and in fact was the first to show that the deep sea (off the continental shelf) had high numbers of species. Following these pioneering studies, one of the earliest systematic studies of marine sediments was the HMS Challenger expedition of 1872–1876, the first global expedition. The reports of the expedition were extensive but were mostly descriptive, relating to taxonomy and general natural history.