bluuebunny/crossref_metadata_2025 huggingface dataset:
Tables again
Authors: Clark Malcolm
Published: November 2023

There is quite another way entirely to tackle tables. Tex has two commands, \halign and \ valign, which provide an enormously flexible way of handling tabular material. With the exception of the two ‘programming’ examples from the last chapter, all the tabular material presented so far would (in my opinion) be more easily handled through this alternative technique. I confess that I very rarely use \settabs. The\halign command in particular seems to lend itself to a variety of problems. To begin with, its use gives the illusion of awkwardness. This is more a function of unfamiliarity, rather than intrinsic difficulty.
Rules
Authors: Clark Malcolm
Published: November 2023

But there is something fairly important missing. What about the rules? We normally expect tables to be boxed in. This we can do, but the alignments tend to get even more hidden in the welter of extra detail. The use of the horizontal rule to divide some rows is still quite common, but vertical rules are now infrequently found. A curious inversion of reality has occurred. The latest edition of the Chicago Manual of Style says One style that has changed since the last edition of this manual was published is the use of vertical rules in tabular matter. In line with a nearly universal trend among scholarly and commercial publishers, the University of Chicago Press has given up vertical rules as a standard feature of tables in the books and journals that it publishes. The handwork necessitated by including vertical rules is costly no matter what mode of com position is used, and in the Press’s view the expense of it can no longer be justified by the additional refinement it brings.
Fonts
Authors: Clark Malcolm
Published: November 2023

In this chapter we will look at some of the aspects of dealing with fonts. To some extent the basics have already been covered. As already related, a simple \font\tenss=cmss10 loads the Computer Modern Sans Serif font at 10 pt. There are potential problems since plain is only guaranteed to set up the basic 16 fonts. These are the eight Computer Modern fonts in Figure 18.1. All are at 10 pt, but four of them are also loaded at some other sizes: A number of others are \preloaded in plain. All that \preloaded means is that the font information is read in so that should we need those fonts sometime there is no overhead.
Making pages
Authors: Clark Malcolm
Published: November 2023

Although we have looked at some aspects of building pages, we have not tackled this in a systematic way. In this chapter we will look at the default arrangements that Tex has for page building, and the ways this may be modified for more specialist requirements. In particular, we shall examine some of the techniques which may be employed in order to achieve double-column output. This seems a perennial problem, not least since the most common size of paper available, generally of the order of just over 8 inches, or about 750 points, requires some alternative to the default strategy when faced with 10 point fonts.
Breaking up
Authors: Clark Malcolm
Published: November 2023

Some topics, although essential, have been treated in a rather diffuse manner. In particular the problem of line, paragraph, and page breaking requires closer examination. It reveals some fascinating aspects of Tex- In order to be able to tackle this, we need to have some knowledge about three kinds of costs associated with this activity. In a nutshell, Tex seeks to minimize some function of the costs incurred in dividing a paragraph into lines. The three costs are termed badness, penalties, and merits (or demerits). Penalties have at least been introduced in a rather loose and informal way. The function of a penalty is to modify Tex’s ‘normal’ or default behaviour. All of these behaviour characteristics can be modified either globally or locally by either modifying global penalty values, or by locally introducing specific penalties. Only penalties may be specified or altered. The other costs are determined by Tex itself. ‘Badness’ is associated with individual lines (or potential lines) within a paragraph, while ‘merit’ is calculated over the whole paragraph. Tex can report their values, but you have no mechanism for altering them directly. The merit figure may be altered indirectly through manipulation of \penalty.
Delays and deferments
Authors: Clark Malcolm
Published: November 2023

The action of page creation has some interesting consequences. There are certain matters which Tex deliberately delays until a page has been created. Clearly, the \output routine makes explicit certain actions, but there are a few implicit actions too. To be specific, the \mark command has several compatriots, namely the triad \topmark, \firstmark, and \botmark whose meaning is determined through page breaking. The contents of these three will change automatically, without any ‘human’ (or even explicit command) intervention. The basic strategy is this: somewhere in the text, there will be some \marked text. The first time that \mark is used, all of the \mark triad take the same value.
Collections
Authors: Clark Malcolm
Published: November 2023

The purpose of this chapter is to draw together some of the commands which have been uncovered in the evolution of the primer. From time to time it has been necessary to introduce members of various classes of commands in a somewhat unstructured way. They were needed to solve particular problems, but elaborating upon them and introducing similar commands would confuse the flow of the text. The account of 1E)C’s primitive conditionals has been rather scattered throughout the preceding chapters. It is useful to draw all the variations into a more ordered exposition, augmenting them with examples where appropriate. Some of the conditionals are used infrequently, although it may be comforting to know that they are available. Naturally these tests would not occur in ‘normal’ text: they would be located within more elaborate commands.
Random Walk On A Lattice
Authors: Hughes Barry d.
Published: November 2023

The problem of wandering at random in a network of streets, or as we would now describe it, of random walk on a lattice, was first raised by Polya (1919, 1921), though its one-dimensional antecedents may be found in the theory of games of chance, especially the problem of gambler’s ruin (Feller 1970); in one dimension, Polya’s problem is also equivalent to the problem of random flight in one dimension, discussed in §2.2.3. Polya addressed in particular the following problem*. A wandering point (or in modern parlance, a random walker) moves at random between the sites of a simple hypercubic lattice in d dimensions, so that the allowed positions of the random walker are points indexed by a d-dimensional vector with integer components. The motion consists of a sequence of steps between nearest-neighbour sites-one of the coordinates of the position changes by ±1 in any step. It is assumed that there is unbiased random motion, so that all of the 2d possible next steps away from the current position are equally likely, each having probability 1/(2d).
Continuous-Time Random Walks
Authors: Hughes Barry d.
Published: November 2023

In Chapter 4, we obtained the governing equations of continuous-time random processes by taking a suitable joint limit Δ -→0 and τ -→0 of a lattice random walk, where Δ is the lattice spacing and τ the constant time between successive steps. The continuous-time processes so obtained are governed by stable densities (§4.3), with the Gaussian density representing the most typical case. In taking the continuum limit the lattice structure and the identity of individual steps are lost, and the analysis of many properties of the processes so obtained requires rather deep and subtle aspects of advanced probability theory, in contrast to the elegant conceptual simplicity of the original random walk. There is a less drastic way to obtain a continuous time process related to the ordinary random walk, without losing either the lattice structure or the notion of well-defined individual steps. This approach, the continuous time random walk model, owes its popularity in contemporary physics to Montroll and Weiss (1965). From a mathematical standpoint it is a particularly useful special case of the semi-Markov process (Levy 1954, Smith 1955, Pyke 1961, Feller 1964, Qinlar 1969, 1975).
Exploration And Trapping
Authors: Hughes Barry d.
Published: November 2023

In Chapter 3 we addressed at some length for lattice random walks questions arising from P6lya’s problem, which may be succinctly described as ‘Will I come back, and if so, when?’ and ‘Where am I likely to be now?’. In the resolution of these questions, we were neither obliged to record or study the time-history of the walk, nor to take note of the portion of the lattice explored by the walker. We did remark in §3.2.6 that for walks which are not strongly transient, very tortuous paths back to the starting site are statistically significant. It may be recalled from the corollary to P6lya’s Theorem in §3.3.3 that an unbiased walk with finite mean-square displacement per step on a periodic lattice of dimension d is strongly transient if and only if d ≥ 4. In Chapter 4 we observed that the set of sites visited may, in some statistical sense, possess a fractal dimension, influenced by the range of the allowed displacements. The geometrical properties of the set of sites visited are important in some contexts. For example, if one places the random flight model of a polymer chain (§2.6) on a lattice (cf. Chapter 7), the geometry of the set of sites visited in a specific walk determines the shape of individual polymer chains.
The Self-Avoiding Walk
Authors: Hughes Barry d.
Published: November 2023

Therandom walk problems discussed in Chapters 2-5 are loosely described by physicists as Markovian, by which they mean the absence of dependence of future evolution on the entire past history of the process, that is, the absence of memory. For example, in P6lya’s problem of a random walk on a lattice, to determine the probability laws governing future motion of the walker one needs to know only the current position; where the walker was three steps ago is of no significance. In the context of continuous-time random walks, we examined in Chapter 5 some walks with a ‘convolution-type memory’, exhibited explicitly in the generalized master equation formalism. Allowing the walker to have such a memory does not destroy the elegant simplicity of the random walk problem. What kind of memory effects have to be introduced to present real difficulties in the analysis? It is evident that a finite-range memory, in which the transition probability distribution at the nth step depends on the k most recent steps (with k < ∞), leads to nothing new. Such a walk may always be re-defined in terms of a walk with internal states (§3.5). The situation is entirely different, however, if one allows the walker to remember where he has been ever since his walk began, with a memory which does not fade, as the following canonical example shows.
Bernoulli Site Percolation
Authors: Hughes Barry D
Published: November 2023

This chapter contains the rigorous proofs of a number of important results of percolation theory. The results are established here for the specific case of Bernoulli site percolation on a lattice for which the coordination number of a site is bounded. Since the Bernoulli bond problem can always be converted to a Bernoulli site problem (on the covering lattice), the discussion effectively covers the Bernoulli bond problem also. Many of the results of this section have analogues for a more general class of models, allowing for directed bonds and for long-range connections between sites, some of which are summarized in §1.5.7 and §1.5.8.
Critical Exponents In Percolation Theory
Authors: Hughes Barry D
Published: November 2023

It is believed that the behaviour of the key statistics of percolation theory close to the percolation threshold can be described in terms of ‘critical exponents’. Some of the critical exponents are defined solely in terms of the cluster-size distribution Pn(p) and functions related to it, such as the mean cluster size x(p) and the percolation probability P00(p). These are discussed for arbitrary homogeneous lattices in §4.1. A number of relations between the critical exponents defined via Pn(P) are believed to hold (§4.2). These relations are closely analogous to scaling relations in statistical mechanics {cf. Volume 1, §7.6) and can be derived by heuristic scaling arguments (§4.2.2 and §4.2.3). One of these relations can be established rigorously as an inequality by a simple argument (§4.2.4), while a number of other rigorous inequalities can be derived via lattice animal expansions (§4.3).
The Ant In The Labyrinth
Authors: Hughes Barry D
Published: November 2023

It is generally believed that the qualitative properties of myopic (Polya) ants and blind ants are similar, but one important difference when the ant is introduced on a finite cluster has been noted by Mitescu and Roussenq (1983): the ‘equilibrium’ probability distributions of the ant, approached asymptotically as the number of steps grows without bound, differ for the two models. The existence of equilibrium probability distributions on finite clusters can be discussed from the point of view of Markov chains (Feller 1970), where such distributions are called ‘invariant measures’, but here we shall confirm their existence by elementary arguments.
Bisectors and Spinal Spheres
Authors: Goldman William m.
Published: November 2023

We call such hypersurfaces spinal spheres. (Mostow calls a bisector a spinal surface.) Giraud [65] and Mostow [128] describe the structure of bisectors in terms of a foliation by complex hyperplanes orthogonal to the spine σ.
Numerical Invariants
Authors: Goldman William m.
Published: November 2023

This chapter summarizes several invariants in complex hyperbolic geometry. First, the “invariante angulaire” discussed by Cartan [21] is presented. This invariant is associated to triples of points on the absolute and has been used by Toledo in his investigation [41, 163, 162] of surface group actions on complex hyperbolic space. Following that is the “complex cross-ratio” introduced by Koranyi and Reimann in [105]. Finally, inspired by these invariants and thedistance formulas, we introduce an invariant associated to a real geodesic and a complex hyperplane, which is used in the following chapter to investigate intersections of bisectors.
Introduction
Authors: Hannabuss Keith
Published: November 2023

In the first quarter of this century physicists were slowly and somewhat reluctantly forced to realize that the view of the physical world, so painstakingly built up over the preceding four centuries, was in serious need of revision. In particular, the laws that governed the motion of particles on the atomic scale were not those of classical mechanics. Physics found itself catapulted into the strange new world of quantum theory in which the very notion of position or velocity of a particle became blurred and statistical laws dominated everything.
Scattering and tunnelling
Authors: Hannabuss Keith
Published: November 2023

Ever since Rutherford’s pioneering work, passing alpha rays through a metal foil target, beams of particles have provided a means of probing the fine details of subatomic structure. Modern accelerators hurl particles through a target at velocities only slightly less than that of light. The wavelength of each particle shrinks as its momentum increases and it becomes possible to distinguish features far beyond the resolving power of any microscope. In the simplest cases (such as Rutherford’s sedate Edwardian experiments) the particles emerge from their encounter unscathed, and it is by analysing the changes to their momentum that one must build up a picture of the target. To develop some feeling for what happens we shall look at the simplest case in which the particle suffers no deflection. We therefore consider particles moving along a line on which there is some kind of potential barrier.
The mathematical structure of quantum theory
Authors: Hannabuss Keith
Published: November 2023

We have already mentioned that Schriöidinger’s formulation of quantum mechanics was slightly preceded by a totally different algebraic formulation proposed by Heisenberg. Although Schriöidinger’s methods were so much simpler that they were generally preferred, both approaches seemed to give the same answers, so it was natural to ask how they were related. This was particularly important because some features of quantum theory were easier to understand in Heisenberg’s formalism. In fact, as we have already noted, Schriöidinger was able to show that the two methods were essentially equivalent. Soon mathematicians like Hermann Weyland John von Neumann were able to find a general mathematical scheme that encompassed both approaches, and it is this that we shall now describe.
The commutation relations
Authors: Hannabuss Keith
Published: November 2023

Our results so far, in particular Section 3.4, strongly suggest that although it is possible to find formalisms in which either momentum or position are easy to handle, one cannot simultaneously manage both. The underlying reason for this is that the operators P and X do not commute.
Variational methods
Authors: Hannabuss Keith
Published: November 2023

Variational methods exploit an alternative characterization of eigenvectors and eigenvalues which lends itself better to approximation schemes. This is motivated by the geometrical fact that the points on a quadric, x.A.x = 1, that are closest to or furthest from the centre lie on the principal axes, and these point along eigenvectors of the matrix, A, defining the quadric.
The semi-classical approximation
Authors: Hannabuss Keith
Published: November 2023

There is another very useful sort of approximation which compares a quantum system not with a simpler quantum system, but with the corresponding classical system.
Relativistic wave equations
Authors: Hannabuss Keith
Published: November 2023

One serious objection to the whole of quantum theory as we have developed it so far, is that it is inconsistent with the theory of relativity. In order to see how we may overcome this defect, at least as far as the special theory of relativity is concerned, we shall first recall some of its basic features. Shortly after Einstein’s original 1905 papers on the subject, Minkowski realized that the physical ideas could also be understood geometrically, by combining space and time into a single four-dimensional real vector space, M, whose elements we shall call four-vectors.
Skew-Symmetric Scalar Products
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

Chapter 2 was concerned with complex spaces having an indefinite scalar product. The detailed study of Riccati equations with real coefficients requires that we also consider scalar products on ℝn generated by real, invertible, skew-symmetric matrices; a further step away from the familiar euclidean situation. Nevertheless, the connections with the indefinite scalar products introduced in Chapter 2 are strong. For example, if K and A are real n x n matrices with K nonsingular and skew-symmetric, A will be said to be skew-symmetric with respect to K if KA is symmetric (see Section 3.2 below). On the other hand, iK is hermitian and because (iK)(iA) is also real and symmetric, it follows that iA is iK-self-adjoint in the sense of Section 2.2. Thus, the theory of Chapter 2 applies as well, but the investigations of this chapter in terms of real matrices yield more detailed information in situations relevant to our subsequent study of real Riccati equations.
Rational Matrix Functions
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

In this chapter we study r x n matrices W(A.) whose elements are rational functions of a complex variable λ. Thus we may write
Gemetric Theory: The Complex Case
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

Using some of the material developed in Part I we can now begin with the first central chapter on solution of Riccati equations. The “geometric theory” of the title refers to the geometry of subspaces in a finite dimensional complex linear space. It is shown that solutions of Riccati equations, and hermitian solutions in particular, are strongly connected with invariant subspaces of a certain operator defined by the coefficients of the equation. By examining these invariant subspaces in the context of a suitable indefinite scalar product (see Chapter 2) considerable information on the structure of the solution set of the Riccati equation is obtained. Of course, the “solution set” here admits both real and complex matrices.
Hermitian Solutions and Factorizations of Rational Matrix Functions
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

Chapters 7 and 8 focus on a geometric theory connecting hermitian solutions of the symmetric algebraic Riccati equation (7.2.1) with invariant subspaces of the associated Hamiltonian matrix . In Chapter 9 a quite different approach is developed using an iterative process. In the present chapter a third, apparently quite different, line of attack is developed. It begins with the easy observation that, if (7.2.1) (or (10.1.1) below) has a hermitian solution X then the rational matrix function
Perturbation Theory
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

In this chapter we present the basic results concerning behaviour of hermitian solutions of algebraic Riccati equations under perturbations, or changes, of the coefficient matrices of the CARE (10.1.1). In the first section we consider the problem of existence of hermitian solutions, while in the later sections the focus is on the behaviour of extremal solutions or, more generally, on unmixed solutions.
Propositional calculus
Authors: Ince D C
Published: November 2023

To describe the nature of propositions and propositional expressions. To describe what mathematical proof involves. To describe how rules of inference can be used in the proof process. To show how propositional calculus can be used during requirements analysis and system specification. The process of system specification and requirements analysis involves examining the behaviour of a proposed software system and asking questions which enable a correct system specification to be developed. For example, the text of a statement of requirements may contain sentences such as If the reactor is on-line and the operator has typed in a query command, then the query log will be updated.
Operations and objects in Z
Authors: Ince D C
Published: November 2023

To describe a large part of the Z toolkit. To describe the use of schemas in the definition of generics. To provide further practice in the use of schemas. The primary aim of this chapter is to look at the facilities offered within the Z language. These facilities which include basic data types such as the natural numbers also include operations on set, function and sequence data types. The facilities are frequently referred to as the Z toolkit.
Zand design
Authors: Ince D C
Published: November 2023

Aims To describe the difference between functional specification and design. To show one use of mathematical proof within formal software development. The previous chapters have described the process whereby a system is specified in terms of mathematics, using a language for specifying stored data, data invariants, pre-conditions, and post-conditions. A point which was made at the beginning of the book was that discrete mathematics is an excellent medium for specifying a system, because it just describes the bare bones of that system. A feature of the specifications that have been presented in the last two chapters is that they do not contain any design directives: the behaviour of the system is specified in terms of operations on mathematical structures, many of which are not implemented in programming languages.
Chronicle completion in 𝒦-IA
Authors: Sandewall Erik
Published: November 2023

As defined in Chapter 2, subsection 2.3.1, a chronicle in 𝒦-IA is a tuple of the form

〈 𝒦, 0, 〉IA, LAW〈SCD, OBS〉

The DFL logic from Chapter 6 provides a formal language for expressing the action laws, the schedule, and the observations. The preSCDt chapter uses the trajectory semantics to define the relationship between a 𝒦-IA chronicle consisting of DFL formulae on one hand, and a set of IDS developments on the other. It applies equally for linear and for branching time.
Probability Density Estimation
Authors: Bishop Christopher M
Published: November 2023

Density estimation can also be applied to unlabelled data (that is data without any class labels) where it has a number of applications. In the context of neural networks it can be applied to the distribution of data in the input space as part of the training process for radial basis function networks (Section 5.9), and to provide a method for validating the outputs of a trained neural network (Bishop, 1994b). In Chapter 6, techniques for density estimation are combined with neural network models to provide a general framework for modelling conditional density functions.
Complete and categorical theories
Authors: Manzano María
Published: November 2023

This chapter is devoted essentially to the study of completeness of theories and the introduction of various methods for verifying this property, which were defined in Chapter 4. As we indicated in Chapter 5, a complete theory syntactically characterized is in itself a decision procedure; every sentence is either deducible or refutable using the theory itself as an oracle. As we will see, this means that the class of models of a complete theory is strongly connected by means of the relation of elementary equivalence.
CoBase: a cooperative database system*
Authors: Chu Wesley W , Chen Q , Merzbacher M
Published: November 2023

This chapter proposes the use of the type abstraction hierarchy (TAH) as a framework for deriving cooperative query answers (CQA). The TAH integrates abstraction with the subsumption (is-a) and composition (part-of) semantics found in the type hierarchy. This framework provides a multi-level object representation that is an important aspect of cooperative query answering. Patterns, which specify one or more conditions on an object, are introduced as a small granularity structure with specific semantic information. Cooperative query answering uses the TAH and patterns to provide query relaxation, generalization, and specialization. Relaxation can be explicitly controlled by the user, implicitly performed by the system, or both. An explanation system is also included to present the relaxation path, as well as a nearness measure between the approximate answer and the exact answer.
On using logic to represent belief
Authors: Nossum Rolf T
Published: November 2023

The overall goal of this chapter is to convey knowledge of some techniques for representing and reasoning about intentional concepts in non-classical logics. Such logics are of central importance in representing and reasoning about belief and knowledge, intention and obligation, planning and acting, and other concepts. Philosophers aptly call them logics of Practical Reasoning. These logics are gaining wide acceptance as representational vehicles for advanced applications in databases, artificial intelligence, intelligent tutoring, and other fields. These logics are gaining wide acceptance as representational vehicles for advanced applications in databases, artificial intelligence, intelligent tutoring, and other fields.
Deriving answer for safty queries
Authors: Demolombe Robert , Jones Andrew
Published: November 2023

In addition to asking questions concerning the actual content of the database, a user may also wish to query the reliability of the messages it contains. We argue that, in reasoning about the reliability of the information in his database ( or its ‘Safety’, as we choose to call it), the database administrator will need to consider the source, or sources, which sent in the signals containing this information. In attempting to answer ‘Safety queries’, therefore, the administrator will need to be able to consider not only what is believed or known to be the case, but also the signalling acts which various agents have performed when supplying input. Here we show how some tools from applied modal logic–specifically, the logic of action, and logics of belief and knowledge–may be used to represent reasoning about Safety. We supply both a general formal analysis of the concepts involved, and a simplified formalization suitable for implementation. This is a preliminary report on work in progress, and so we conclude with a list of problems that might be investigated within this framework.
A modal analysis of subjunctive queries
Authors: Boutilier Craig
Published: November 2023

An important and well-studied problem in philosophical logic, artificial intelligence, and database systems is that of modelling theory change or belief revision. That is, given a knowledge base KB, we want to characterize semantically the knowledge base that results after learning a new fact α. However, the question of how to revise KB is important not just in the case of changing information, but also when we want to investigate questions of the form ‘What if A were true?’ A subjunctive conditional A > B is one of the form‘If A were the case then B would be true.’ Subjunctives have been widely studied in philosophy and it is generally accepted that (some variant of) the Ramsey test is adequate for evaluating the truth of such conditionals
A Theory and its Metatheory in FS
0
Authors: Matthews Seán
Published: November 2023

Feferman, in the previous chapter, has proposed FS, a theory of finitary inductive systems, as a framework theory that allows a user to reason both in and about an encoded theory. I look here at how practical FS really is. To this end I formalise a sequent calculus presentation of classical propositional logic, and show this can be used for work in both the theory and the metatheory. The latter is illustrated with a discussion of a proof of Gentzen’s Hauptsatz.
A Unified View of Consequence Relation, Belief Revision, and Conditional Logic *
Authors: Katsuno Hirofumi , Satoh Ken
Published: November 2023

The notion of minimality is proving to be a key unifying idea in three different areas of Artificial Intelligence: non-monotonic reasoning, belief revision, and conditional reasoning. However, it is difficult for the readers of the literature in these areas to perceive the similarities clearly. The models used in each area differ, sometimes superficially and sometimes in depth, and the notation is different. This makes it hard to apply results on, say, conditional logic to, say, belief revision. Even within the same area there is confusion, as, for example, different authors use different formalisms for conditional logic, sometimes without relating their proposals to the literature. We therefore present a uniform view of how minimality is used in these three areas, shedding light on deep connections among the areas. We clarify differences and similarities between different approaches by classifying them according to the notion of minimality on which they are based.
Later Analysis and the Infinite
Authors: Russ Steve
Published: November 2023

The final group of works translated here, RZ, F, F+, and PU date from the last two decades of Bolzano’s life and the earliest of them (RZ and F) were written about fifteen years after the publication of his early analysis works BL, RB, and DP. Those years had been a momentous middle period of Bolzano’s life. He had been deposed from his university post in 1819 and forced to retreat from public life. He turned this to advantage by composing his fourvolume Wissenschaftslehre where he develops the conceptual and philosophical framework that is necessary for his views and contributions in the fields of logic and the theory of knowledge.
Introduction
Authors: Kangshen Shen , Crossley John N , Lun Anthony W -C
Published: November 2023

The book is anonymous like many Chinese classics. The ancient tradition was for successive writers to modify the original text over many generations and, even when a work is ascribed to an author, the latest text may contain contributions from a number of later writers.
Liu Hui’s Preface to his Running Commentary on the Nine Chapters
Authors: Kangshen Shen , Crossley John N , Lun Anthony W -C
Published: November 2023

This very brief and terse preface gives us all the information we have about Liu Hui the individual. From the History of the Sui Dynasty we are able to infer that Liu was alive in 263 AD.
Distribution by Proportion
Authors: Kangshen Shen , Crossley John N , Lun Anthony W -C
Published: November 2023

The 20 Problems in this Chapter mainly concern proportional distribution, including direct proportion, inverse proportion and compound proportion. So this Chapter can be regarded as a continued development of the Rule of Three from Chapter 2. The Problems may be classified into three groups.
Vegard’s law in random two-dimensional alloys
Authors: Thorpe M F , Jin, W , Mahanti S D
Published: November 2023

In all the work on the vibrational spectra of disordered alloys, much less attention has been given to changes in bond lengths, than has been given to changes in masses and force constants. However, length changes are often important in determining both local and global properties. In this paper, we focus on length changes in the simplest non-trivial systems. We have found it convenient to set up an effective medium theory, of the kind that Roger Elliott has exploited so effectively in many problems. In the end we find simple qualitative explanations for our results that have direct and obvious experimental consequences.
Raman scattering from multilayers
Authors: Barrio Rafael A
Published: November 2023

The Raman effect is a complicated interaction of light with matter in which photons, electrons, and phonons play an equally important role. The experiment itself is very simple. One illuminates a sample with a beam of monochromatic light of fixed frequency and detects the intensity of light emanating from the sample at a certain angle and with different frequencies. The difference between the frequency of the incident and of the scattered light (Raman shift) is due to the creation or annihilation of phonons in the sample; therefore, a Raman spectrum gives information about the normal modes of vibration of the atoms in the material. The rigorous description of the Raman effect is not a simple task, since a good knowledge of the electronic bands and of the electron-phonon interaction is needed. Nevertheless, Raman scattering has become a familiar and useful technique to characterize molecules and perfect crystals because symmetry imposes severe selection rules due to momentum conservation.
Exactly solvable problem for the polar optical phonon Green’s function and the phonon distribution function in the vibrational spectrum diffusion region of uniaxial crystal solid solutions
Authors: Jpatova P , Shchukin V A
Published: November 2023

The electron energy spectrum of three-dimensional disordered systems such as mixed crystals is known to contain a diffusion region situated between the continuous spectrum formed by extended electron states and the density-of-states tail created by localized electron states. The electron mean free path l in the diffusion region is of the order of magnitude of the electron wavelength A, and the electron motion is controlled by collisions. There is no small parameter in the diffusion region of three-dimensional disordered systems; therefore both: (1) the electron multiple scattering by isolated fluctuations of composition; and (2) the interference of scattering processes from different fluctuations are of the same order of magnitude.
Spin fluctuations in antiferromagnets and superconductors
Authors: Fleury Paul A
Published: November 2023

Until the early 1960s, inelastic light scattering was confined to the study of vibrational degrees of freedom in condensed matter through the familiar Raman effect. With the invention of the laser and the development of improved photon detectors and spectrometers, the utility of inelastic light scattering for the study of essentially all types of elementary excitations in condensed matter rapidly advanced. A key element in advance was a short paper in 1963 by Roger Elliott and Rodney Loudon which first raised the possibility of what they called ‘the electronic Raman effect’. This paper also carried a prediction that inelastic light scattering could be used to study one-magnon excitations in antiferromagnets.
A new approach to the calculation of spectra for strongly correlated systems
Authors: Costi T A , Hewson A C
Published: November 2023

Direct diagonalization of a small cluster embedded within a larger system gives some information about the larger system but only over the largest energy scale. Such cluster calculations may help to identify peaks in XPS and other spectra in this energy range, but are of little direct use in giving information about the lower energy excitations and thermodynamic properties. For these lower energy excitations larger range correlations have to be taken into account. Whatever the computing resources, there is a rather sharp cut-off on the size of the cluster that can be diagonalized numerically. There are various possibilities for extending these to obtain results on smaller energy scales and larger clusters.
One-dimensional magnets and conformal invariance
Authors: Ziman Timothy
Published: November 2023

One-dimensional magnetism began as a testing ground for theoretical models but has become a subject of experimental investigation as well. Nowadays we know a number of insulating materials that provide good realizations of one-dimensional antiferromagnets of various quantum spin S: CuC122NC5H5, which has spin S = ½; CsNiCl3 and Ni(C2H8N2)iNO2(ClO4), both with S = 1; CsVCl3 (S = i); and (CH3  )4NC5H5MnCl3, which has S = i (Steiner et al. 1976; Renard et al. 1988). To a first approximation the magnetic ions of each material form well separated chains each of which can be described by the very simple Hamiltonian.
The t-1
4 universality class: theory and simulation of slow domain growth after quenches to low and finite temperatures
Authors: Lindgard Per-Anker , Castan Teresa
Published: November 2023

The kinetics of domain growth is of relevance for the formation of polycrystalline microstructures. This is of considerable importance in surface science (Wang and Lu 1983), metallurgy (Hashimoto et al. 1976), and earth science (Salje 1988). It is, in practice, the complexity of the microstructures, which is decisive for the physical properties.
Melting curves of two- and three-dimensional Wigner electron crystals in a magnetic field
Authors: Lea M J , March N H
Published: November 2023

It is a pleasure to present this work on Wigner electron crystallization in an applied magnetic field in this volume honouring Roger Elliott. It is now more than two decades since one of us (NHM), his then research student, John Durkan, and Professor Elliott, presented work at the San Francisco Conference on the Metal-Insulator Transition, on the way in which Wigner electron crystallization could be aided by the application of a magnetic field (Durkan et al. 1968).
Quantum optical properties of the exciton--polariton: new families of excitonic quasiparticles
Authors: Birman Joseph L
Published: November 2023

In this chapter I want to present some highlights of our group’s recent research on novel quantum optical properties of the familiar exciton-polariton and a few results on new composite quasiparticles related to excitons in semiconductors. This family of quasiparticles includes the familiar exciton-polariton which comprises exciton plus coupled photon as well as two extensions of the polariton: the phonoriton, invented by L. V. Keldysh and collaborators, which is a composite of exciton and photon and phonon; and the photoriton, a parametrically created composite of two off-resonance photons and the exciton-this was also introduced by the L. V. Keldysh group.
Albrecht Durer
Authors: Coolidge Julian Lowell , Gray Jeremy
Published: November 2023

It is a noteworthy fact in geometry that the study of figures in three or more dimensions has lagged very far behind that of figures which are in one plane. We have already mentioned on p. 12 that this was a source of regret to Plato. There was really a deep underlying reason for this, even if Plato were unaware of the fact. It can be shown that the geometry of the plane is not only simpler than that of higher spaces, but actually richer in interesting results. Such matters are not, however, our present concern; historically the lag of solid geometry behind plane geometry has persisted from Plato’s time to our own.
Afterword (1992): a revolution in the historiography of mathematics?
Authors: Crowe Michael
Published: November 2023

The revolution has begun! The concluding words in my 1975 paper, which appears as Chapter 1 of this volume, were that whether or not revolutions happen in mathematics, revolutions can ‘occur in mathematical nomenclature, symbolism, metamathematics (e.g. the metaphysics of mathematics), methodology (e.g. standards of rigour), and perhaps even in the historiography of mathematics’. No more compelling evidence that a revolution in the historiography of mathematics is under way could be found than the essays that Donald Gilles has collected for inclusion in this volume. Written by a distinguished group of historians and philosophers from a variety of countries, these papers collectively demonstrate that new and sophisticated approaches to the history of mathematics are being employed with success. Never before has this claim seemed so justified.
Vortices And Fluids: From Gravity To Elasticity
Authors: Meli Domenico Bertoloni
Published: November 2023

In the second half of the seventeenth century natural philosophy relied heavily on Cartesian explanations of phenomena in terms of the motion of particles of different shape and size. Even if the specific mechanism devised by Descartes was questioned, vortices and subtle fluids were the cornerstone for an intelligible explanation of the universe. The following two sections examine some aspects of the vortex theories framed by Descartes and Huygens in order to gain some familiarity with this material. The examples I have selected were well known to Leibniz and Newton, and were important in the formation of their views; we shall find several references to them in Leibniz’s manuscripts.
The Private Itinerary
Authors: Meli Domenico Bertoloni
Published: November 2023

This chapter outlines the itinerary leading from the Notes, namely Leibniz’s working sheets containing his immediate thoughts on the Principia, to the Tentamen. All the manuscripts discussed here are unpublished: those marking the most significant stages in Leibniz’s itinerary are reproduced in the Appendix, where I also provide a detailed textual analysis. In this chapter, however, I present a broader picture, including some manuscripts which are not reproduced later and the relevant Marginalia. Indeed, in some cases Leibniz’s isolated annotations in his own copy of Newton’s masterpiece can only be understood in relation to the texts presented here.
Reflections On Leibniz’s Theory And Its Development
Authors: Meli Domenico Bertoloni
Published: November 2023

The previous two chapters have charted the formation and development of Leibniz’s theory from his reading of the Principia, through a series of manuscript essays and calculations, to the published Tentamen. A comparative study of the material presented so far and of further private manuscripts, letters, and public texts discussed in this chapter allows us to grasp the problems met by Leibniz. Taking into account the contrast between these different genres, and the style of writing in the Tentamen, it is possible to appreciate the development of his theory and strategy. The following section examines the structure and style of the Tentamen, emphasizing the difference between the genesis of Leibniz’s ideas and the order of presentation. I also select two examples relevant to the relationships between mathematical theory and physical interpretation, namely the generalization of the expression for centripetal force attained in De Conatu, and the law of the harmonic circulation. My contention is that in the first case the link between mathematics and physics was lacking and Leibniz’s result was therefore unsuitable for a response to Newton. The second example, however, in which mathematics and physics are brought together, appeared in print in a prominent position.
The proposal
Authors: Fowler D H
Published: November 2023

Plato’s Meno is an enquiry into virtue. As part of the discussion, Plato’s Socrates digresses to illustrate his opinion that we cannot teach anything, but can only prompt recollection of the opinions of a disembodied soul. But Socrates’ digression takes him into the subject of our enquiry here, so let us begin with this passage, Meno 8le-85d. The translation is by W. K. C. Guthrie; I have suppressed all editorial additions, made one very slight alteration throughout which will be explained in Section 1.2(c), below, and restored two speeches at the end of 83b that have unaccountably been omitted.
PLato’s Mathematics Curriculum In Republic Vii
Authors: Fowler D H
Published: November 2023

My attitude to Plato and the Academy has been greatly influenced by the sceptical and rigorous analysis described in Cherniss, The Riddle of the Early Academy, especially Lectures 1 and 3 therein. And, to a historian of ancient mathematics, an additional riddle is raised by Cherniss’s scrupulous exploration of the fragmentary evidence, in the tension between his conclusion, on the one hand, that: Philodemus says that mathematics made great progress under the direction of Plato, who formulated problems which the mathematicians zealously investigated.
Elements Iv, X, And Xiii: The Circumdiameter And Side
Authors: Fowler D H
Published: November 2023

Let us now turn to the problem of the circumdiameter and side, introduced by my Socrates at S41 It is very similar in spirit to the very fruitful problem of the diagonal and side, discussed in Section 2.2, out of which grew the remarkable discoveries of the dimension of squares in Chapter 3. We take a regular polygon with its circumscribed circle, and try to describe the relation between the circumdiameter and a side. Instead of the side we can take a diagonal; and instead of the diameter, we can take the radius, since doubling or halving a ratio has an effect on the anthyphairesis that seems difficult to elucidate. We can also consider the inscribed circle, or investigate the similar three-dimensional problems associated with an edge or diagonal of a regular polyhedron inscribed in or circumscribed around a sphere.
The Nature Of Our Evidence
Authors: Fowler D H
Published: November 2023

Plato, it is said, placed an inscription over the door to the Academy: “Let no one unskilled in geometry enter”. Since my aim is to propose a new and central role for Plato in the development of early Greek mathe matics, this well-known and often-cited story provides an epitome of the main theme of this book.
Numbers And Fractions
Authors: Fowler D H
Published: November 2023

Almost the only notation for numbers (i.e. numerals) found in the earliest and best manuscripts of Euclid’s works (apart from the Sectio Canonis) is used to label the propositions; with the exception of four numerals in Elements XIII 11, those few numbers that are used in the text are written out as words. This is an extreme case; most mathematical texts, and all manner of financial, administrative, and legal documents, employ a notation that will underly the main topic of this chapter.
Later Interpretations
Authors: Fowler D H
Published: November 2023

It has been an oft-repeated story since antiquity that the origins of Greek geometry are to be found in Egyptian land measurement; for a discussion of some of the texts, see Heath, HGM i, 121-8. The earliest such possible reference comes from Herodotus, Histories ii, 109, written in the fifth century BC: This king [the semi-mythical Sesostris] moreover (so they said) divided the country among all the Egyptians by giving each an equal square parcel of land, and made this his source of revenue, appointing the payment of a yearly tax.