bluuebunny/crossref_metadata_2025 huggingface dataset:
An intuitionistic theory of types
Authors: Martin-Löf Per
Published: November 2020

The theory of types with which we shall be concerned is intended to be a full scale system for formalizing intuitionistic mathematics as developed, for example, in the book by Bishop 1967. The language of the theory is richer than the language of first order predicate logic. This makes it possible to strengthen the axioms for existence and disjunction. In the case of existence, the possibility of strengthening the usual elimination rule seems first to have been indicated by Howard 1969, whose proposed axioms are special cases of the existential elimination rule of the present theory. Furthermore, there is a reflection principle which links the generation of objects and types and plays somewhat the same role for the present theory as does the replacement axiom for Zermelo-Fraenkel set theory. An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis. Mathematical objects and their types. We shall think of mathematical objects or constructions. Every mathematical object is of a certain kind or type. Better, a mathematical object is always given together with its type, that is, it is not just an object, it is an object of a certain type.
The origins of societies
Authors: Maynard Smith John , Szathmary Eors
Published: November 2020

The heat generated within a mound of the termite Macrotermes is carried upwards by a central air duct. The air then travels down along narrow channels close to the surface of the mound, where it is cooled, and where, as in a lung, oxygen and carbon dioxide are exchanged. The whole mound is an airconditioning system. Although the mound resembles a human building in having features ensuring the comfort of its inhabitants, it differs in that not one of its builders had a picture of the completed structure before building started. The structure emerged from the rule-governed behaviour of tens of thousands of interacting workers. In this, the mound resembles a human body rather than a human building. The body is built by the rule-governed actions of millions of cells. Nowhere is there anything resembling a blueprint of the body. At most, the genome is a set of instructions for making a body: it is not a description of a body. The resemblance between the development of an insect colony and of an organism has led to the concept of a ‘superorganism’. The analogy has some value. To the extent that individual ants, bees or termites have lost the capacity to reproduce, they can propagate their genes only by ensuring the success of the colony, just as somatic cells can propagate theirs only by ensuring the success of the organism. Hence, the colony can be expected to have features adapted to ensure its success, and it is reasonable to apply concepts of optimization to it, rather than to the individual—as was done, for example, by Oster & Wilson (1978) in their book on insect caste systems. But for our purposes the concept of a super organism is of little use. To understand the origins of animal societies, we must ask how individuals capable of reproduction came to cooperate to the extent that most of them lost the ability to reproduce. To understand their maintenance, we must explain why they are not disrupted by cheating. Unlike somatic cells, the individual workers, although related, are not genetically identical. We would therefore expect within-colony conflict to be widespread, as indeed it is: examples discussed below concern egg laying by workers, and conflict over the sex ratio.
The origin of language
Authors: Maynard Smith John , Szathmary Eors
Published: November 2020

The past 30 years has witnessed a debate between the holders of two very different views about how humans are able to talk. The behaviourists, following B. F. Skinner, argue that we learn to talk in the same way that we learn any other skill. Children are rewarded when they speak correctly, and reproved when they make mistakes. We can talk, whereas chimpanzees cannot, because we are better at learning: there is nothing special about language. In contrast, Noam Chomsky and his followers have argued that humans have a peculiar competence for language, which is not merely an aspect of their general intelligence. We learn to utter, and to understand, an indefinitely large number of grammatical sentences, and to avoid an even larger number of ungrammatical ones, so we cannot possibly learn which sentences are grammatical by trial and error. Instead, we must learn the rules that generate grammatical sentences. These rules are of great subtlety, so that, although we acquire and apply them, we cannot formulate them explicitly. For example, consider the two following sentences: How do you know who he saw? (1) Who do you know how he saw? (2) How do you know who he saw? Who do you know how he saw? Every speaker of English knows at once that is grammatical, and is not. But what rule tells us this? No-one but a trained linguist would have any idea, any more than a non-biologist would know how the rate of beating of the heart is adjusted to meet changing demands. In section 17.3, we describe a hypothesis about the rule that tells us that is ungrammatical: it is a subtle rule, but as yet no-one has thought up a simpler one. It is hard to believe that we could so painlessly master such rules unless we were genetically predisposed to do so. More generally, it is still beyond the wit of linguists and computer scientists to write a language-translating programme, yet many 5-year-olds know two languages, do not mix them up, and can translate from one to the other. A second reason for thinking that we cannot learn to talk by trial and error lies in the poverty of the input on which a child must rely. After hearing a finite set of utterances, a child learns to generate an indefinitely large number of grammatical sentences. This implies that the child learns rules, and not merely a set of sentences.
New methodologies in the preparation of imprinted polymers
Authors: Alexander Cameron , Kirsch Nicole
Published: November 2020

Molecular imprinting is a rapidly emerging method for the creation of recognition sites in synthetic polymers, and the resultant materials offer considerable promise as selective adsorbents in a number of applications. The technique exploits the principle of using elements of a target molecule to create its own recognition site. This is achieved by the formation of a highly cross-linked polymeric matrix around a template, which can be the target molecule itself or a close structural analogue. The key to this procedure is to ensure that, during the polymerization, functional groups of the template molecule are fully engaged in interactions with ‘complementary functionality’ of polymer-forming components. These interactions are then ‘locked in’ by the incorporation of the whole assembly into the polymer structure. Subsequent removal of the template reveals the newly created binding sites containing functional groups in the precise stereochemical arrangement to ensure recognition of the target in a highly selective manner (Scheme 1). The first reports of molecular imprinting in organic polymers involved the templating of protected sugars, in the form of esters with a polymerizable boronic acid (however, see Ref. 18 for an earlier example of the imprinting concept) into a cross-linked polymer ‘scaffold’, and variations of the basic technique have now been adopted by many research groups around the world. In general, molecularly imprinted polymers (MIPs) are prepared by thermal or photochemical free-radical routes, employing acrylic or vinylic monomers in a solvent chosen to ensure that the final matrix is microporous. The numbers and types of molecules which have now been imprinted is very large, but a key factor in the preparation of MIP materials with the desired recognition properties is still the chemical nature of the link between the template and the polymer backbone. Consequently, strategies by which the template can be securely fixed in space as the growing matrix forms around it, yet be readily removed to generate the recognition site after polymer synthesis is complete, are of particular interest.
The Bayes Paradigm, Estimation and Information Measures
Authors: Grenander Ulf , Miller Michael I.
Published: November 2020

The basic paradigm is the Bayesian setup, given is the source of parameters X ∈ X which are seen through a noisy channel giving observations Y ∈ Y. The posterior distribution determines the bounds on estimation of X given Y, the risk associated with estimating it, as well as a characterization of the information in the observation in Y about X.
Gaussian Random Fields on Undirected Graphs
Authors: Grenander Ulf , Miller Michael I.
Published: November 2020

This chapter focuses on state spaces of the continuum studying Gaussian random fields on discrete lattices. Covariances are induced via difference operators and the associated neighborhood structure of the resulting random field is explored. Texture representation and segmentation are studied via the general Gaussian random field structures. For dealing with the partition function determined by the log-determinant of the covariance asymptotics are derived connecting the eigenvalues of the finite covariance fields to the spectrum of the infinite stationary extension.
Second Order and Gaussian Fields
Authors: Grenander Ulf , Miller Michael I.
Published: November 2020

This chapter studies second order and Gaussian fields on the background spaces which are the continuum limits of the finite graphs. For this random processes in Hilbert spaces are examined. Orthogonal expansions such as Karhunen–Loeve are examined, with spectral representations of the processes established. Gaussian processes induced by differential operators representing physical processes in the world are studied.
Time and classification
Authors: Kemp T.S.
Published: November 2020

The age and the classification of a particular fossil are the two fundamental properties necessary to begun understanding how it fits into the evolutionary patterns revealed by the fossil record. There are often misunderstandings of one or other of these by specialists. Evolutionary biologists on occasion express far too optimistic a view of how accurately fossils can actually be dated, both absolutely and relative to one another. Geologists have been known to have a rather limited view of how modern systematic methods are used to infer relationships from large amounts of information, be it morphological or molecular. In this chapter, a brief outline of the principles underlying the construction of the geological timescale, and of a classification are given, along with reference timescales and classifications for use throughout the following chapters. The creation of a timescale for dating the events recorded in the rocks since the origin of the Earth is one of the greatest achievements of science, unspectacular and taken for granted as it may often be. It is also unfinished business insofar as there are varying degrees of uncertainty and inaccuracy about the dates of many rock exposures, none more so than among the mostly continental, rather than marine sediments containing the fossils with which this work is concerned. A geological timescale is actually a compilation of the results of two kinds of study. One is recognising the temporal sequence of the rocks, and agreeing arbitrarily defined boundaries between the named rock units, the result of which is a chronostratic timescale. The other is calibration of the sequence and its divisions in absolute time units of years before present, a chronometric timescale. It is simple in principle to list the relative temporal order of events, such as the occurrence of fossils, in a single rock unit, although even here the possibility of missing segments, known as hiatuses, in local parts of the unit, or of complex folding movements of the strata disturbing the order must not be forgotten. The biggest problem is correlating relative dates between different units in different parts of the world.
Evolution of mammalian biology
Authors: Kemp T.S.
Published: November 2020

There are large biological differences between the mammals and the primitive living amniotes as represented by turtles, lizards, and crocodiles ● Differentiated dentition with occluding post-canine teeth, and radical reorganisation of jaw musculature to operate them ● Differentiation of vertebral column and limb musculature, and repositioning of limbs to bring feet under the body, increasing agility of locomotion ● Relatively huge brain and highly sensitive sense organs ● Endothermic temperature physiology, with very high metabolic rates, insulation, and high respiratory rates ● Precise osmoregulatory and chemoregulatory abilities using loops of Henle in the kidney and an array of endocrine mechanisms Incomplete as it is, the fossil record of the mammal-like reptiles, or ‘non-mammalian synapsids’ permits the reconstruction of a series of hypothetical intermediate stages that offers considerable insight into how, when, and where this remarkable transition occurred. Deriving these stages starts with a cladogram of the relevant fossils that is then read as an evolutionary tree, with hypothetical ancestors represented by the nodes. The characters that define a node, plus the characters of the previous nodes, constitute the reconstruction. The differences in characters between adjacent nodes represent the evolutionary transitions that by inference occurred, and the whole set of successive nodes generates all that can be inferred about the sequence of acquisition of characters. If a hypothetical ancestral synapsid is placed at the base of the cladogram, and a hypothetical ancestral mammal as the final node, then the set of nodes in between represents everything the fossil record is capable of revealing about the pattern by which mammalian characters evolved: the sequence of their acquisition, the correlations between characters, and possibly the rates of their evolution. Of course, the inferred pattern of evolution of characters is only as reliable as the cladogram which generated it, and that in turn is only as realistic as the model of evolution used in its construction from the character data. And of course, there must have been many intermediate stages in the transition than cannot be reconstructed for want of appropriate fossil representation of those particular grades. Nevertheless, limited as it may be, this is what can be known from the fossil record.
Applications
Authors: Petersen Wesley , Arbenz Peter
Published: November 2020

Linear algebra is often the kernel of most numerical computations. It deals with vectors and matrices and simple operations like addition and multiplication on these objects. Vectors are one-dimensional arrays of say n real or complex numbers x0, x1, . . . , xn−1. We denote such a vector by x and think of it as a column vector, On a sequential computer, these numbers occupy n consecutive memory locations. This is also true, at least conceptually, on a shared memory multiprocessor computer. On distributed memory multicomputers, the primary issue is how to distribute vectors on the memory of the processors involved in the computation. Matrices are two-dimensional arrays of the form The n · m real (complex) matrix elements aij are stored in n · m (respectively 2 · n ·m if complex datatype is available) consecutive memory locations. This is achieved by either stacking the columns on top of each other or by appending row after row. The former is called column-major, the latter row-major order. The actual procedure depends on the programming language. In Fortran, matrices are stored in column-major order, in C in row-major order. There is no principal difference, but for writing efficient programs one has to respect how matrices are laid out. To be consistent with the libraries that we will use that are mostly written in Fortran, we will explicitly program in column-major order. Thus, the matrix element aij of the m×n matrix A is located i+j · m memory locations after a00. Therefore, in our C codes we will write a[i+j*m]. Notice that there is no such simple procedure for determining the memory location of an element of a sparse matrix. In Section 2.3, we outline data descriptors to handle sparse matrices. In this and later chapters we deal with one of the simplest operations one wants to do with vectors and matrices: the so-called saxpy operation (2.3). In Tables 2.1 and 2.2 are listed some of the acronyms and conventions for the basic linear algebra subprograms discussed in this book.
Algorithms and Turing Machines
Authors: Penrose Roger , Gardner Martin
Published: November 2020

What Precisely is an algorithm, or a Turing machine, or a universal Turing machine? Why should these concepts be so central to the modern view of what could constitute a ‘thinking device’? Are there any absolute limitations to what an algorithm could in principle achieve? In order to address these questions adequately, we shall need to examine the idea of an algorithm and of Turing machines in some detail. In the various discussions which follow, I shall sometimes need to refer to mathematical expressions. I appreciate that some readers may be put off by such things, or perhaps find them intimidating. If you are such a reader, I ask your indulgence, and recommend that you follow the advice I have given in my ‘Note to the reader’ on p. viii! The arguments given here do not require mathematical knowledge beyond that of elementary school, but to follow them in detail, some serious thought would be required. In fact, most of the descriptions are quite explicit, and a good understanding can be obtained by following the details. But much can also be gained even if one simply skims over the arguments in order to obtain merely their flavour. If, on the other hand, you are an expert, I again ask your indulgence. I suspect that it may still be worth your while to look through what I have to say, and there may indeed be a thing or two to catch your interest. The word ‘algorithm’ comes from the name of the ninth century Persian mathematician Abu Ja’far Mohammed ibn Mûsâ alKhowârizm who wrote an influential mathematical textbook, in about 825 AD, entitled ‘Kitab al-jabr wa’l-muqabala’. The way that the name ‘algorithm’ has now come to be spelt, rather than the earlier and more accurate ‘algorism’, seems to have been due to an association with the word ‘arithmetic’. (It is noteworthy, also, that the word ‘algebra’ comes from the Arabic ‘al-jabr’ appearing in the title of his book.) Instances of algorithms were, however, known very much earlier than al-Khowârizm’s book.
Truth, Proof, and Insight
Authors: Penrose Roger , Gardner Martin
Published: November 2020

What is truth? How do we form our judgements as to what is true and what is untrue about the world? Are we simply following some algorithm - no doubt favoured over other less effective possible algorithms by the powerful process of natural selection? Or might there be some other, possibly non-algorithmic, route - perhaps intuition, instinct, or insight - to the divining of truth? This seems a difficult question. Our judgements depend upon complicated interconnected combinations of sense-data, reasoning, and guesswork. Moreover, in many worldly situations there may not be general agreement about what is actually true and what is false. To simplify the question, let us consider only mathematical truth. How do we form our judgements - perhaps even our ‘certain’ knowledge - concerning mathematical questions? Here, at least, things should be more clear-cut. There should be no question as to what actually is true and what actually is false - or should there? What, indeed, is mathematical truth? The question of mathematical truth is a very old one, dating back to the times of the early Greek philosophers and mathematicians - and, no doubt, earlier. However, some very great clarifications and startling new insights have been obtained just over the past hundred years, or so. It is these new developments that we shall try to understand. The issues are quite fundamental, and they touch upon the very question of whether our thinking processes can indeed be entirely algorithmic in nature. It is important for us that we come to terms with them. In the late nineteenth century, mathematics had made great strides, partly because of the development of more and more powerful methods of mathematical proof. (David Hilbert and Georg Cantor, whom we have encountered before, and the great French mathematician Henri Poincaré, whom we shall encounter later, were three who were in the forefront of these developments.) Accordingly, mathematicians had been gaining confidence in the use of such powerful methods. Many of these methods involved the consideration of sets with infinite numbers of members, and proofs were often successful for the very reason that it was possible to consider such sets as actual ‘things’ - completed existing wholes, with more than a mere potential existence.
Where Lies The Physics of Mind?
Authors: Penrose Roger , Gardner Martin
Published: November 2020

In discussions of the mind-body problem, there are two separate issues on which attention is commonly focused: ‘How is it that a material object (a brain) can actually evoke consciousness?’; and, conversely; ‘How is it that a consciousness, by the action of its will, can actually influence the (apparently physically determined) motion of material objects?’ These are the passive and active aspects of the mind-body problem. It appears that we have, in ‘mind’ (or, rather, in ‘consciousness’), a non-material ‘thing’ that is, on the one hand, evoked by the material world and, on the other, can influence it. However, I shall prefer, in my preliminary discussions in this last chapter, to consider a somewhat different and perhaps more scientific question - which has relevance to both the active and passive problems - in the hope that our attempts at an answer may move us a little way towards an improved understanding of these age-old fundamental conundrums of philosophy. My question is: ‘What selective advantage does a consciousness confer on those who actually possess it?’ There are several implicit assumptions involved in phrasing the question in this way. First, there is the belief that consciousness is actually a scientifically describable ‘thing’. There is the assumption that this ‘thing’ actually ‘does something’ - and, moreover, that what it does is helpful to the creature possessing it, so that an otherwise equivalent creature, but without consciousness, would behave in some less effective way. On the other hand, one might believe that consciousness is merely a passive concomitant of the possession of a sufficiently elaborate control system and does not, in itself, actually ‘do’ anything. (This last would presumably be the view of the strong-AI supporters, for example.) Alternatively, perhaps there is some divine or mysterious purpose for the phenomenon of consciousness - possibly a teleological one not yet revealed to us - and any discussion of this phenomenon in terms merely of the ideas of natural selection would miss this ‘purpose’ completely.
Transformation operators for symmetry elements
Authors: Newnham Robert E.
Published: November 2020

Since the physical property tensors depend on symmetry, mathematical methods for determining the influence of symmetry are needed. More specifically, transformation matrices are required for the symmetry elements that generate the crystal class. As explained earlier, some classes require only one symmetry element while others may require two or three. None require more than three. This chapter also includes a brief discussion of the seven Curie groups used for textured solids and liquids. As explained in Chapter 2, transformations from one coordinate system to another can be specified by a set of nine direction cosines, aij where i, j = 1, 2, 3. The first index, i, refers to the “new” or transformed axis, the second index j to the “old” or reference axis. There are four types of symmetry elements that require discussion: rotation axes, mirror planes, inversion centers, and inversion axes in which rotation is accompanied by inversion. All rotations are assumed to be in the counterclockwise direction. Other symmetry elements such as rotoreflection axes are not needed. The fourteen symmetry elements in Table 4.1 generate the 32 crystal classes. No proof is offered here but this statement can be verified geometrically using the stereographic projections in Chapter 3. There are two final points concerning these transformation matrices. First, keep in mind that they must obey the orthogonality conditions described in Chapter 2. This provides a useful way of avoiding mistakes. The second point concerns handedness. Symmetry elements involving reflection or inversion reverses the handedness of the coordinate system. This of course includes inversion axes such as 1 ̅̅, 2 ̅ = m, 3 ̅, 4 ̅, and 6 ̅. The handedness change can be verified by showing that the determinant of the transformation matrix is −1. For ordinary rotation axes such as 2, 3, 4, and 6, there is no change in handedness, and the determinant is +1. In Table 4.2 we list the minimum symmetry requirements for each of the 32 crystal classes. These are the transformation operations needed to develop the physical property matrices for single crystals. These techniques will be described in the next chapter.
Stress and strain
Authors: Newnham Robert E.
Published: November 2020

Stress (force per unit area) and strain (change in length per unit length) are both symmetric second rank tensors like the dielectric constant, but they are not property tensors. Experimenters are at liberty to apply different types of forces to a specimen, therefore there is no reason that the stress tensors (and the resulting strain tensor) must conform to the crystal symmetry. Stress and strain tensors do not obey Neumann’s Law. They are sometimes called field tensors to distinguish them from property tensors like the dielectric constant. Property tensors are relationships between field tensors. For the same reason, electric and magnetic fields are first rank field tensors, as are magnetization and polarization. They do not obey the symmetry principles as first rank property tensors such as pyroelectricity or the magnetocaloric effect are required to do. In arbitrary coordinate systems, the state of stress in a specimen is described by nine components of the stress tensor: The first subscript refers to the direction of the force, the second to the normal to the face on which the force acts. To prevent translational motion, each force is balanced by an equal and opposite force on the reverse side of the specimen. Stress component X22 is a tensile stress in which both the force and the normal are along Z2, and X12 is a shear stress in which a force along Z1 acts on a face normal to Z2. For static equilibrium, the torques must be balanced, otherwise rotation occurs; this means that the stress tensor must be symmetric with X12 = X21, X13 = X31, and X23 = X32. Thus the stress state is specified by six independent components: three tensile stresses X11, X22, and X33, and three shear components X12, X13, and X23. For an arbitrary axial system (new axes) the general stress tensor can be rewritten as a 6 × 1 column matrix: The first three components in the column matrix are tensile stresses along Z'1, Z '2, Z '3, and the last three are shear stresses about Z '1, Z '2, Z '3. Both the tensor and matrix forms are widely used in the literature.
Elasticity
Authors: Newnham Robert E.
Published: November 2020

All solids change shape under mechanical force. Under small stresses, the strain x is related to stress X by Hooke’s Law (x) = (s)(X), or the converse relationship (X) = (c)(x). The elastic compliance coefficients (s) are generally reported in units of m2/N, and the stiffness coefficients (c) in N/m2. For a fairly stiff material like a metal or a ceramic, c is about 1011 N/m2 = 1012 dynes/cm2 = 100 GPa = 0.145 × 108 PSI. Hooke’s Law is a linear relation between stress and strain, and does not describe the elastic behavior at high stress levels that requires higher order elastic constants (Chapter 14). Irreversible phenomena such as plasticity and fracture occur at still higher stress levels. Two directions are needed to specify stress (the direction of the force and the normal to the face on which the force acts), and two directions are needed to specify strain (the direction of the displacement and the orientation of the measurement axis). Thus there are four directions involved in measuring elastic stiffness, which is therefore a fourth rank tensor: . . . Xij = cijklxkl . . . .
Nonlinear phenomena
Authors: Newnham Robert E.
Published: November 2020

The physical properties discussed thus far are linear relationships between two measured quantities. This is only an approximation to the truth, and often not a very good approximation, especially for materials near a phase transformation. A more accurate description can be obtained by introducing higher order coefficients. To illustrate nonlinearity we discuss electrostriction, magnetostriction, and higher order elastic, and dielectric effects. These phenomena are described in terms of fourth and sixth rank tensors. Many of the recent innovations in the field of electroceramics have exploited the nonlinearities of material properties with factors such as electric field, mechanical stress, temperature, or frequency. The nonlinear dielectric behavior of ferroelectric ceramics (Fig. 15.1), for example, has opened up new markets in electronics and communications. In these materials the electric polarization saturates under high fields. Electric displacement Di varies with applied electric field Ej as . . . Di = εijEj + εijkEjEk + εijklEjEkEl +· · · , . . . where εij is the dielectric permittivity and εijk and εijkl are higher order terms. The data in Fig. 15.1 were collected for a relaxor ferroelectric in its paraelectric state above Tc where the symmetry is centrosymmetric. Therefore the third rank tensor εijk is zero, and the shape of the curve is largely controlled by the first and third terms. For cubic crystals, the fourth rank tensor εijkl is similar in form to the elastic constants discussed in Chapter 13. Tunable microwave devices utilize nonlinear dielectrics in which the polarization saturates as in Fig. 15.1. By applying a DC bias the dielectric constant can be adjusted over a wide range.
Evidence for catastrophic organic changes in the geological record
Authors: Hallam Tony
Published: November 2020

If asked what they understood by the word ‘catastrophe’, most people would probably agree that it was something big, bad, and sudden, and involved damage to organisms. In the natural world today, perhaps the most striking catastrophes result from major earthquakes, in which thousands of people can be killed within minutes. Going back through human history, we allow for greater stretches of time. Thus, in the middle of the fourteenth century, over a period of five years, an estimated one-third of the European population died directly as a result of catching the plague: the ‘Black Death’. By any reckoning this ranks as a catastrophe. It had a dramatic effect on European society for many years. When we extend our consideration to geological time, in which it is routine to deal with changes taking place over millions of years, events lasting only a few thousand years may be regarded as catastrophic if the contrast with the ‘background’ is sharp enough. Various definitions have been proposed for a mass extinction. A conveniently concise if imprecise one that I favour is that it is the extinction of a significant proportion of the world’s living animal and plant life (the biota) in a geologically insignificant period of time. The imprecision about the extent of an extinction can be dealt with fairly satisfactorily in particular instances by giving percentages of fossil families, genera, or species, but the imprecision about time is more difficult to deal with. An important question about mass extinctions is to assess how catastrophic they were, so we also require a definition of ‘catastrophe’ in this context. One thought-provoking attempt at such a definition is that a catastrophe is a perturbation of the biosphere that appears to be instantaneous when viewed at the level of detail that can be resolved in the geological record. At this point more needs to be said about the nature of the geological record. The material that geologists and palaeontologists deal with occurs in the layered successions of sedimentary rocks, mainly sandstones, shales, and limestones, that can clearly be observed in good rock exposures, either natural ones, as in coastal cliffs or mountains, or artificial ones, as in quarries or borehole cores.
Pulling the strands together
Authors: Hallam Tony
Published: November 2020

In drawing together the various strands we first need to ask how catastrophic, as opposed to merely calamitous, the various mass-extinction events were. As was indicated in Chapter 3, there is no way in which the stratigraphic record can ever provide dates that are precise to within less than a few thousand years. Thus, the connection between a bolide impact and a catastrophic phase of extinction lasting no longer than a few years could never be established with a high degree of confidence from the record of the strata alone. All that can be done is to establish a pattern that is consistent with such a scenario. As was also pointed out in Chapter 3, a change that is drastic enough over an interval of a few thousand to a few tens of thousands of years can reasonably be described as catastrophic in the context of normal patterns of geological change extending over millions of years. Several events seem to qualify unequivocally: the end-Permian, the end-Cretaceous, and, on a smaller scale, the end-Palaeocene, which affected only one group of deep-sea organisms. It needs to be added, though, that the end-Cretaceous event seems to have been the culmination of a phase of increased extinction rates among a wide variety of organisms. Such patterns of catastrophic change cannot yet be ruled out for the other mass-extinction events, but decisive evidence is not yet forthcoming. A more gradual or multiple pattern of extinctions appears to be more likely for the end-Ordovician, late Devonian, and end-Triassic extinctions and also for more minor ones such as those in the early Jurassic and mid-Cretaceous. Catastrophic coups de grâce are quite possible, if not probable, as culminating factors for some of these events, but more detailed collecting and statistical work across the world is required to put forward a stronger case than has been made so far. It has been claimed that the ‘big five’ mass extinctions are something special, as opposed to lesser extinction events, in so far as they were too drastic and rapid in their effects on many organisms to give time for normal Darwinian adaptive responses to operate.
Computer Simulations of Dislocations
Authors: Bulatov Vasily , Cai Wei
Published: November 2020

This book presents a broad collection of models and computational methods - from atomistic to continuum - applied to crystal dislocations. Its purpose is to help students and researchers in computational materials sciences to acquire practical knowledge of relevant simulation methods. Because their behavior spans multiple length and time scales, crystal dislocations present a common ground for an in-depth discussion of a variety of computational approaches, including their relative strengths, weaknesses and inter-connections. The details of the covered methods are presented in the form of "numerical recipes" and illustrated by case studies. A suite of simulation codes and data files is made available on the book's website to help the reader "to learn-by-doing" through solving the exercise problems offered in the book.
Interfacial phenomena
Authors: Bokstein Boris S. , Mendelev Mikhail I. , Srolovitz David J.
Published: November 2020

An interface is a surface across which the phase changes. Interfaces must be present in all heterogeneous systems, such as those discussed above. Interfacial properties necessarily differ from those of the bulk phases since the atomic bonding/structure of an interface represents a compromise between those of the phases on either side of the interface. For example, an atom at a free surface, which is an interface between a condensed phase and a gas (or a vacuum), generally has fewer neighbors with which to bond than it would have if it were in the bulk, condensed phase. In an equilibrium multi-component system, the chemical potential of each species must be the same in all phases, as well as at the interface. Not surprisingly, the chemical composition of the interface will, in general, differ from that of the bulk. For example, molecules in a gas (or solute in a condensed phase) can adsorb (segregate) onto the surface (interface) of a condensed phase. Interfacial processes play important roles in all areas of materials science and in many (most) areas of modern technology. As the trend toward miniaturization in microelectronics continues and interest in nanoscale structures grows, interfacial phenomena will become even more important. Clearly, the ratio of the number of atoms at surfaces and interfaces to those in the bulk grows as system size decreases (70% of the atoms in a nanometer diameter particle are on a surface!). Therefore, the thermodynamic properties of a system become increasingly dominated by interfacial properties as the dimensions of the system shrink. We can distinguish several types of interfaces: solid–liquid, liquid–gas, solid–gas, solid phase α–solid phase β, and grain boundaries. The meaning of the first four types of interface is self-explanatory. Grain boundaries represent a special class of interfaces; interfaces across which the phase does not change. What does change abruptly across this interface is the spatial orientation of the crystallographic axes. Most crystalline materials are polycrystalline, which means that they are composed of a large number of grains, each with a unique crystallographic orientation with respect to some laboratory frame of reference.
Introduction To Liquids
Authors: Nitzan Abraham
Published: November 2020

The statistical mechanics of atomic motion in gases and solids have convenient starting points. For gases it is the ideal gas limit where intermolecular interactions are disregarded. In solids, the equilibrium structure is pre-determined, the dynamics at normal temperature is characterized by small amplitude motions about this structure and the starting point for the description of such motions is the harmonic approximation that makes it possible to describe the system in terms of noninteracting normal modes (phonons). Liquids are considerably more difficult to describe on the atomic/molecular level: their densities are of the same order as those of the corresponding solids, however, they lack symmetry and rigidity and, with time, their particles execute large-scale motions. Expansion about a noninteracting particle picture is therefore not an option for liquids. On the other hand, with the exclusion of low molecular mass liquids such as hydrogen and helium, and of liquid metals where some properties are dominated by the conduction electrons, classical mechanics usually provides a reasonable approximation for liquids at and above room temperature. For such systems concepts from probability theory (see Section 1.1.1) will be seen to be quite useful. This chapter introduces the reader to basic concepts in the theory of classical liquids. It should be emphasized that the theory itself is general and can be applied to classical solids and gases as well, as exemplified by the derivation of the virial expansion is Section 5.6 below. We shall limit ourselves only to concepts and methods needed for the rest of our discussion of dynamical processes in such environments.
Introduction To Stochastic Processes
Authors: Nitzan Abraham
Published: November 2020

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(t) are made at discrete time 0 < t1 < t2, . . . ,< tn < T. In this case the sequence {z(ti)} is a discrete sample of the stochastic process z(t). Let us start with an example. Consider a stretch of highway between two intersections, and let the variable of interest be the number of cars within this road segment at any given time, N(t). This number is obviously a random function of time whose properties can be deduced from observation and also from experience and intuition. First, this function takes positive integer values but this is of no significance: we could redefine N → N − (N) and the new variable will assume both positive and negative values. Second and more significantly, this function is characterized by several timescales: 1. Let τ1 is the average time it takes a car to go through this road segment, for example 1 min, and compare N(t) and N(t +∆ t) for ∆ t << τ1 and ∆ t τ1. Obviously N(t) ≈ N(t + ∆ t) for ∆ t << τ1 while in the opposite case the random numbers N(t) and N(t + ∆ t) will be almost uncorrelated. Figure 7.1 shows a typical result of one observation of this kind. The apparent lack of correlations between successive points in this data set expresses the fact that numbers sampled at intervals equal to or longer than the time it takes to traverse the given distance are not correlated. 2. The apparent lack of systematic component in the time series displayed here reflects only a relatively short-time behavior.
The Quantum Mechanical Density Operator And Its Time Evolution: Quantum Dynamics Using The Quantum Liouville Equation
Authors: Nitzan Abraham
Published: November 2020

The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
Computable functions and semicomputable sets on many-sorted algebras
Authors: Tucker J. V. , Zucker J. I.
Published: November 2020

The theory of the computable functions is a mathematical theory of total and partial functions of the form f : Nn →N, and sets of the form. . . SÍ Nn. . .that can be defined by means of algorithms on the set . . . N = {0,1,2, . . . } . . . of natural numbers. The theory establishes what can and cannot be computed in an explicit way using finitely many simple operations on numbers. The set of naturals and a selection of these simple operations together form an algebra. A mathematical objective of the theory is to develop, analyse and compare a variety of models of computation and formal systems for defining functions over a range of algebras of natural numbers. Computability theory on N is of importance in science because it establishes the scope and limits of digital computation. The numbers are realised as concrete symbolic objects and the operations on the numbers can be carried out explicitly, in finitely many concrete symbolic steps. More generally, the numbers can be used to represent or code any form of discrete data. However, the question arises: . . . Can we develop theories of functions that can be defined by means of algorithms on other sets of data?. . . The obvious examples of numerical data are the integer, rational, real and complex numbers; and associated with these numbers there are data such as matrices, polynomials, power series and various types of functions. In addition, there are geometric objects that are represented using the real and complex numbers, including algebraic curves and manifolds. Examples of syntactic data are finite and infinite strings, terms, formulae, trees and graphs. For each set of data there are many choices for a collection of operations from which to build algorithms. . .How specific to the set of data and chosen operations are these computability theories? What properties do the computability theories over different sets of data have in common? . . . The theory of the computable functions on N is stable, rich and useful; will the theory of computable functions on the sets of real and complex numbers, and the other data sets also be so? The theory of computable functions on arbitrary many-sorted algebras will answer these questions.
Equational Logic Programming
Authors: O’Donnell Michael J.
Published: November 2020

Sections 2.3.4 and 2.3.5 of the chapter ‘Introduction: Logic and Logic Programming Languages’ are crucial prerequisites to this chapter. I summarize their relevance below, but do not repeat their content. Logic programming languages in general are those that compute by deriving semantic consequences of given formulae in order to answer questions. In equational logic programming languages, the formulae are all equations expressing postulated properties of certain functions, and the questions ask for equivalent normal forms for given terms. Section 2.3.4 of the ‘Introduction . . .’ chapter gives definitions of the models of equational logic, the semantic consequence relation . . . T |=≐(t1 ≐ t2) . . . (t1 ≐ t2 is a semantic consequence of the set T of equations, see Definition 2.3.14), and the question answering relation . . . (norm t1,…,ti : t) ?- ≐ (t ≐ s) . . . (t ≐ s asserts the equality of t to the normal form s, which contains no instances of t1, . . . , ti, see Definition 2.3.16).