bluuebunny/crossref_metadata_2025 huggingface dataset:
Logarithms and exponentials
Authors: Reutenauer Christophe
Published: November 2023

The various characterizations of Lie polynomials obtained in Chapter 1 are extended to Lie series, in a straightforward way. These results become really interesting when one considers exponentials of Lie series. This leads naturally to the Campbell–Baker–Hausdorff formula. Section 3.2 is devoted to the canonical projections of the free associative algebra, especially the first one, whose image is the free Lie algebra and which is obtained by a logarithm. Section 3.3 is devoted to the computation of the coefficients of the Hausdorff series. A generalization of this series leads again to the canonical projection. The computation is done using descent numbers of permutations and shuffle algebra. Several properties of the coefficients are established: symmetries, generating functions, and recursions. The last section presents the original methods of Campbell, Baker, and Hausdorff, which explains the introduction of Bernoulli numbers. A closed formula is also given for the Hausdorff series.
Shuffle Algebra and Subwords
Authors: Reutenauer Christophe
Published: November 2023

The shuffle algebra is a free commutative algebra over the set of Lyndon words; this result is presented in Section 6.1, together with a precise identity on the shuffie product of Lyndon words, which implies that actually there is a canonical structure of algebra of divided powers. In Section 6.2 a remarkable presentation of the shuffle algebra is given; the generators are the nonempty words, and the relations the nontrivial shuffle products. In Section 6.3 we introduce subword functions on the free group, the Magnus transformation of the free group, the algebra structure on the module of subword functions, and the fact that this algebra is generated by the particular subword functions corresponding to Lyndon words. The main tools arc the concept of representative, or recognizable functions on the free group, and the infiltration product of Chen, Fox and Lyndon. Section 6.4 presents the commutator calculus of P. Hall, and its generalizations. There are many results involving the lower central series of the free group, the Magnus transformation, and the algebra of subword functions.
Introduction
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

During recent decades technological progress has completely changed the observational methods in all fields of geosciences and geoengineering: the trend is to achieve immediate results, thus reducing time and costs. Modern high speed computers and satellite based techniques are entering more and more disciplines such as cartography, climate investigation, geodesy, geology, geophysics, meteorology, navigation, oceanography, and many others.
Approximate Integration
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

The method of expanding a function into a Fourier series in terms of trial functions (for example, spherical harmonics, certain types of radial basis functions) amounts to the problem of evaluating the expansion coefficients by integration. In practical applications (e.g. in many problems of geodesy or geophysics), however, the integrand is often not known continuously over all of n. Instead we are confronted with the situation that experimental data or measured values yielding the function only at prescribed points must be integrated. This gives rise to approximate integration. The aim of this chapter is to present in more detail some procedures of numerical integration on the sphere. The first one is a low discrepancy method (Section 7.1). It is based on the idea of equidistribution. The conceptual background is a spherical version of the Koksma-Hlawka formula corresponding to invariant pseudodifferential operators. Because of their simple structure and flexibility low discrepancy methods can be recommended for problems involving a large number of densely spaced data points, and under this assumption the method is satisfactory.
Vector Spherical Harmonics
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

Vector spherical harmonics (as well as tensor spherical harmonics) occur in many areas of theoretical physics and engineering. In particular, they are used in quantum mechanics, or whenever one has to deal with vector fields of any geophysical nature. Important examples are gravitational fields, magnetic fields, electromagnetic fields, or elastic fields (for deformation analysis of the earth’s surface using vector spherical harmonics see for example A. BEN-MENAHEM, S.J. SINGH (1981), E.W. GRAFAREND (1982, 1986), J. BRUD0WSKY et al.(1992), w. FREEDEN et al.(1996), and M. TUCKS (1996)). Moreover, in all geosciences there is also a growing need for dealing with vector and tensor fields in spherical geometry, as more and more measurements of vectorial and tensorial type become available by modern space techniques such as satellite-to-satellite tracking or satellite gradiometry.
Abbreviating proofs using metamathematical rules
Authors: Hájek Petr , Montagna Franco , Pudlá Pavel
Published: November 2023

There are examples of sentences which are provable in theory T but which have only very long proofs. All the known examples have the property that at the same time there is a simple meta mathematical argument which shows that they are provable; thus, they have short proofs in some stronger theory whose soundness is as acceptable as the soundness of the original one. Such a phenomenon, which one calls speed up by provability, was first studied by Parikh. He proved in [Par71] that, if T is a recursively enumerable extension of Peano Arithmetic PA, then there is a sentence φ such that the formalization of “ φ is provable in T” has a much shorter proof than φ. This result is generalized in [DM89] and in (Mon90].
Sums over Poisson processes
Authors: Kingman J F C
Published: November 2023

where f is a real-valued function on the state space of a Poisson process IT. There are a number of simple, but very important, results which give conditions for the (absolute) convergence of such sums, and expressions for the expectation, variance, and characteristic function of the random variable L. In this section we shall derive the form of these results without rigorous proof, the latter being deferred to the next section.
Poisson Processes on the line
Authors: Kingman J F C
Published: November 2023

We have been studiously avoiding the most important example of the state space S, the whole real line IR or a subinterval of it. This has been done to stress that most of the general results have nothing to do with the special structure of IR and are equally applicable to Poisson processes in the plane, in 3-space, or in more general spaces S like those we shall meet in Chapter 7. The time has now come to link this general theory with the peculiar properties of IR, notably its order structure.
Curves
Authors: Gurtin Morton E
Published: November 2023

A curve is a set 𝒾 in ℝ together with a smooth map p↦ r(p) from an interval of ℝ into ℝ such that:

𝒾 is the range of r;

5 rPnever vanishes;

the domain of r is either all of ℝ or a bounded interval [P, Q];

if the domain is ℝ, either r is periodic,6 or | r(p) | → ∞ as IPI → ∞.

The function r is a parametrization of 𝒾; the variable p is the parameter; the interval [P, Q] (or ℝ) is the parameter interval; r(P) and r(Q) are the initial and terminal points (or collectively, the endpoints) of 𝒾, and we say that 𝒾 is from r(P) to r(Q).
Energetics and the Dissipation Inequality
Authors: Gurtin Morton E
Published: November 2023

We consider a purely mechanical theory in which the interface is driven by an energy difference between bulk phases. We assume that the body is immobile, incapable of deformation, with energy uniform in each of the bulk phases.
Digression: Statical Theory of Interfacial Stability; Convexity, The Frank Diagram, and Corners; Wulff Regions
Authors: Gurtin Morton E
Published: November 2023

It is a basic physical premise that stable configurations of a mechanical system minimize energy. In this section we will discuss the ramifications of this premise for the phase interface. Aside from being important in its own right, the discussion will furnish a derivation of the (statical) balance relations for capillary forces as Euler – Lagrange equations corresponding to stationary values of the energy, and will provide a physical basis for certain assumptions we will make concerning the interfacial energy.
Evolution Equations for the Interface; Basic Assumptions
Authors: Gurtin Morton E
Published: November 2023

To avoid repeated hypotheses, we will assume throughout the discussion of the mechanical theory that the following hypotheses are satisfied:

(8A) Assumptions

The constitutive equations are compatible with thermodynamics.
Global Behaviour for an Interface with Stable Energy
Authors: Gurtin Morton E
Published: November 2023

I use the term polygyny to denote the peaceful coexistence of two or more mated egg-layers (queens) in the same nest site. Thus I will not consider a number of variants in reproductive structure, as reviewed by Keller and Vargo (this volume).
Constitutive Theory
Authors: Gurtin Morton E
Published: November 2023

16.1 Constitutive equations for the bulk material

We consider, for each of the two phases (i = 1, 2), bulk constitutive equations

with g = grad T the temperature gradient. These yield an analogous relation for the bulk free energy:
Free-Boundary Problems
Authors: Gurtin Morton E
Published: November 2023

The equations derived thus far combine to form an important free-boundary problem for the temperature. The differential equation to be satisfied in bulk is balance of energy supplemented by the constitutive equations (cf. (14.27)): [eq]The evolving interface forms a free boundary, and the corresponding free-boundary conditions follow from the force and energy balances for the interface.
Grid generation as a discrete optimisation problem
Authors: Farmer C L
Published: November 2023

Numerical grid generation involves the problems of:

Boundary fitted coordinates—deforming a regular Cartesian grid to fit complicated boundaries, some of which may be internal.

Scattered data interpolation—deforming a grid so that a fixed set of points lie at grid cell centres or coincide with grid points.

Function approximation with adjustable nodes—deforming a grid so that a given function is equi-distributed.

Methods based on the solution of partial differential equations, variational principles and the direct minimisation of discrete mesh functions have been tried. These techniques generally require a good starting guess and are prone to grid tangling.
Practical flow simulation employing high order accuracy, implicit, finite element based algorithms
Authors: Hickmott S , Hutton A G , Rabbitt M , Smith R M
Published: November 2023

The complexity of most fluid dynamic phenomena is such that, traditionally, practical CFD solvers have adopted fairly simple representations in space and time, embedded in explicit numerical algorithms, in order to render practical problems tractable on available computers. These technologies have been refined to considerable power and sophistication. In the main, they are identified with monotonic finite difference operators written on structured grids (i.e. simple connectivity) facilitating the use of iterative solvers requiring only moderate computer resources.
Physical and numerical aspects of non-equilibrium flows
Authors: Pandolfi Maurizio
Published: November 2023

We look at non-equilibrium phenomena that occur throughout unsteady flows of a compressible, inviscid and non-conductive fluid. The fluid dynamics is then governed by the Euler equations. For the sake of simplicity, and without loss of generality, the flow is assumed to be one-dimensional in the following analysis.
Adaptive finite elements for transonic flows
Authors: Göhner U , Warnecke G , Wendland W
Published: November 2023

An adaptive finite element method for the calculation of transonic potential flows was developed. The adaptivity criteria include residual based error indicators as well as a shock indicator. For a good shock resolution mesh refinement, h-method, as well as moving nodes, r-method, were needed.
Boundary fitted grid generation using tensor product B-splines
Authors: Saunders Bonita V
Published: November 2023

In recent years grid generation, the numerical development of curvilinear coordinate systems, has been instrumental in the development of finite difference and finite element methods for solving partial differential equations (pdes) on arbitrarily shaped domains. In addition to computational aero-dynamics, grid generation is applicable to many diverse areas including hydrodynamics, metallurgy, electromagnetism and other areas involving the solution of pdes over a physical domain.
Riemann solvers: linearizations and adaption
Authors: Toro E F
Published: November 2023

Riemann-problem based methods are one of the most significant contributions to the algorithm-development side of CFD in the last decade. Compared to traditional methods, they are however more complex as well as more demanding on processing power. The solution of the local Riemann problem lies at the centre of both of these disadvantages. Linearized Riemann solvers (eg. Roe, 1981; Botta & Pandolfi, 1989) meant a significant step forward in simplifying algorithms and reducing cost. Practical applications however have shown that the gains may have been at the price of loosing robustness and accuracy.
Algebraic Semantics
Authors: Wagner Eric G
Published: November 2023

The term ‘algebraic semantics’ as used in this chapter, refers to certain approaches to the study of iteration and recursion using either universal algebra or the categorical approach to universal algebra based on Lawvere’s ‘algebraic theories’. The aim of this work is to provide a unified theory of iteration and recursion which covers the common properties of all examples of iteration and recursion. Thus, as we will show, algebraic semantics provides a uniform treatment of the theory of such seemingly diverse subjects as context-free grammars, flowcharts, recursion schemata, and recursively defined domains.
Non-Monotonic Coding Of The Declarative Past
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

Mrs Smith is running a babysitter service. She has a list of reliable teenagers who can take on a babysitting job. A customer interested in a babysitter would call Mrs Smith and give the date on which the babysitter is needed. Mrs Smith then calls a teenager employee of hers and arranges the job. She may need to call several of her teenagers until she finds one who accepts. The customer pays Mrs Smith and the service pays the teenager. The rate is 10 per night unless the job requires overtime (after midnight) in which case it jumps to 15.
A Logical View Of Temporal Database Dynamics
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

In traditional databases, the update of a piece of data means the replacement of the current value of the data by a new one. In temporal databases we are presented with the extra possibility of changing the past, the present and the future. In a nutshell, a temporal update is a ‘change in history’. Note the double reference to time in such an expression: change relates to temporal evolution, while history refers to temporal record. These two notions of time are independent and coexistent. In analysing updates in temporal databases we have to be able to cope simultaneously with those two notions of time. For that, we present next a two-dimensional temporal logic, a formalism that will allow for the simultaneous handling of two references of time.
Temporal Conceptual-Level Databases
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

In this chapter we describe the temporal database concepts from a higher abstraction level. Such an abstraction level is called the conceptual level and it is from this level that the process of designing databases and database applications is normally initiated.
Adding a temporal dimension to a logic system
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by amalgamating L with a pure propositional temporal logic T (such as linear temporal logic with ‘Since’ and ‘Until’) in a special way. We refer to this method as ‘adding a temporal dimension to L’ or just ‘temporalizing L’.We show that the logic system T(L) preserves several properties of the original temporal logic such as soundness, completeness, separation, and decidability.
Post Completeness
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

A logic Lis said to bem-reditcibleif, for every formulaφ (q1, ... , qn) ∉ L,there exist formulasψ1 (p1, … , Pm), … , ψn (p1, … , Pm)such that A logic is calledreducibleif it is m-reducible for somem<w.
Concrete process algebra
Authors: Winskel Glynn , Nielsen Mogens
Published: November 2023

Concurrency theory is the branch of computer science that studies the mathematics of concurrent or distributed systems. In concurrency theory, the design of such mathematics is studied and issues concerning the specification and verification of such systems are analysed. Often, a concurrent system is called a process. In order to analyse a large and complex process it is desirable to be able to describe it in terms of smaller and simpler processes. Thus, it seems natural to have some simple processes—the ones that are not subject to further investigation—and operators on them to compose larger ones thus resulting in an algebraic structure. In order to reason about large processes it is often useful to have a set of basic identities between processes at one’s disposal. The most relevant identities among them are normally called axioms. The axiomatic and algebraic point of view on concurrency theory is widely known as process algebra.
Introduction
Authors: Clark Malcolm
Published: November 2023

Tex predates the current concern with microcomputers. This is both a boon and a bane. Its origins lie, in part, in a series of books, The Art of Computer Programming, written by Donald Knuth. This is a projected seven-volume work, as yet incomplete. The first volume was published in 1968, to be followed by Volume 2 in 1969, and Volume 3 in 1973. As might be inferred from the title of the series, the books contain an appreciable amount of mathematics, of one sort or another.
Further rules
Authors: Clark Malcolm
Published: November 2023

The next example (Figure 16.1) is deceptively simple. What it introduces are two quite important topics. The ‘lesser’ is that it is relatively straightforward to align on a decimal point, simply by treating the integer part as one column and the fractional part (including the decimal) as another column. There are possible refinements on this so that the decimal point itself is always included, but this is inappropriate in this example.
More detailed fonts
Authors: Clark Malcolm
Published: November 2023

There is an area which we have hardly touched, that of the position of characters in a font. The characters in a font may be accessed in a number of ways. We are accustomed to the ‘normal’ way, by actually specifying the character - a, b, c, similarly,some characters are accessed through commands Since there are 128 characters in Computer Modern, it is convenient to divide the table into groups of eight or 16. There is a convenient way to refer to octal (base 8) or hexadecimal (base 16) numbers, rather than decimal numbers. For example, the following are equivalent: \char' 152 and \char"6A. An octal number is preceded by a ', while a hexadecimal number is preceded by a 11 Note that the hexadecimal numbers use the upper-case alphabetic characters. Using lower-case will lead to trouble.
Last words
Authors: Clark Malcolm
Published: November 2023

What will the future of Tex be? Will there be a Tex4 to follow the Tex3 of 1989? It is clear from the statements made by Knuth that he is unwilling to consider this possibility. It may not be impossible, since the change to Tex3 seemed impossible immediately before the Stanford conference at which Knuth announced his intention to make one last major change (the ability for Tex to handle 8-bit characters rather than 7-bit), and a number of less significant (though sometimes extremely useful) changes. Nevertheless, Knuth has said that Tex3 is ‘frozen’, or at least, the only changes which will be made are corrections to the code.
Zelmanov’s Theorem
Authors: Vaughan-Lee Michael
Published: November 2023

In 1989, E. I. Zelmanov [57-59] solved the restricted Burnside problem by proving the following:

Zelmanov’s Theorem  Let L be a Lie algebra over a field K, and let L be generated by a finite set {a1, a2, ... , a,}.
Extors in Projective Space
Authors: Goldman William m.
Published: November 2023

The last two chapters deal with the general theory of bisector intersections. This chapter discusses a class of real hypersurfaces (called “extors”) in complex projective space which extend and generalize bisectors in complex hyperbolic and elliptic geometry. The complex projective theory of extors is simpler than that of bisectors in the metric geometries.
Intersections of Bisectors
Authors: Goldman William m.
Published: November 2023

In real hyperbolic space, bisectors are totally geodesic and so are their intersections. In particular bisector intersections are necessarily connected. However, in complex hyperbolic geometry, this is no longer true. We have already discussed two cases of bisector intersections: cospinal pairs (where the spines lie in a common complex geodesic, see §5.3.1) and comeridianal pairs (where the spines lie in a common meridian, see §5.3.5). Cospinal families of bisectors arose from the orthogonal projections onto complex geodesics. Comeridianal pairs arose in the discussion of Cartan’s configuration of seven ℝ-circles characterizing triples of points lying on a chain.
Measurements and paradoxes
Authors: Hannabuss Keith
Published: November 2023

We noted in Section 6.5 that quantum measurements change the system which is measured. The abrupt change of state caused by a measurement is quite different from the steady evolution described by Schrodinger’s equation, and this shows up particularly clearly if we imagine carrying out frequent repeated measurements on an evolving system.
Systems of several particles
Authors: Hannabuss Keith
Published: November 2023

One often wants to study systems, such as the helium atom of Sections 12.3 and 14.2, that contain more than one particle. We describe two-particle systems by a wave function, ψ (r1, r2), where r1 is the position of the first particle, and r2 that of the second. If we interchange the two particles, then the wave function becomes ψ (r2, r1).

(16.1)
Dirac particles in electromagnetic fields
Authors: Hannabuss Keith
Published: November 2023

So far we have considered only free particles, moving relativistically with out any external forces. However, since there are reasons to expect that electrons can be described by the Dirac equation, we wish to know how they are affected by an electromagnetic field.
Indefinite Scalar Products
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

Our geometric notions of orthogonality and angle extend to ∈n and more general spaces with the aid of a (standard) scalar product. A more general concept has many useful applications, but requires us to abandon some of that geometric intuition. We have in mind the indefinite scalar products of the present chapter. In particular, they play vital parts in the analysis of Riccati equations with the symmetries among the coefficients that arise so frequently in practice. Furthermore, the analysis of Riccati equations with complex coefficients, or with real coefficients, demand careful preparations in the context of complex and real spaces, respectively.
Matrix Theory and Control
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

The ideas and theory presented in this chapter will play important parts in our study of Riccati equations. They revolve around the concepts of controllable (and observable) matrix pairs. The importance of these concepts was recognized in the early years of development of control theory and it was in that context that the terms “controllable” and “observable” arose and have some physical meaning. Our approach uses linear algebra in the development of these concepts because it is this understanding that is required in the sequel. Their significance in the context of systems theory and control is mentioned as the chapter progresses, but is not given complete mathematical development.
H∞ Control Problems
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

Clearly, the minimum of (1) coincides with the greatest lower bound of the set of positive numbers γ for which (2) is satisfied for some admissible feedback K. Control problems in the form (2) are the subject of this chapter. In recent years, H∞ control problems in various guises have become the subjects of intensive research, mainly by electrical engineers. Hundreds of research papers (and some books) using various approaches to the subject have appeared. We mention the sources Francis (1987), and Kaashoek and Ran (1991), though it cannot be claimed that they cover more than a fraction of the significant work in the area. Other important sources are mentioned in the notes of Section 20.4. It turns out that one successful approach to H00 control problems involves algebraic Riccati equations of continuous type. Here, we treat two simple cases of the so-called “standard problem” of H∞ control, and use the Riccati equation as the main tool for their solution.
Contractive Rational Matrix Functions
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

In this chapter we study strict contractions. In particular, realizations of strict contractions are characterized, and their minimal unitary completions are described. It will be seen that a continuous algebraic Riccati equation (of the type studied in Part II) and its hermitian solutions play a crucial role in the analysis of strict contractions.
The Matrix Sign Function
Authors: Lancaster Peter , Rodman Leiba
Published: November 2023

It has been shown in Chapter 7 that (under suitable hypotheses) the hermitian solutions of this Riccati equation are intimately connected to certain n dimensional invariant subspaces of M and, in particular, to the spectral sub space M+ of M corresponding to eigenvalues in the open right half-plane (see Theorem 7.6.2, for example).
Predicate calculus
Authors: Ince D C
Published: November 2023

To show how predicate calculus can represent statements about col¬ lections of objects. To describe some rules of inference in the predicate calculus. To show how predicate calculus can be used for program specification. In the previous chapter propositional calculus was described. System specifications and statements of requirements were used as concrete examples of its utility. It was stressed that the examples used were artificial in the sense that they were small. However, the examples were artificial in another respect. They were carefully chosen so that they could be expressed in propositional calculus. Unfortunately, the propositional calculus alone is an inadequate medium for expressing many system specifications.
Set theory
Authors: Ince D C
Published: November 2023

To introduce the idea of a set as a collection of objects. To describe the important set operators. To outline how sets and predicate calculus can be combined to produce mathematical specifications. To describe some rules of inference involving sets. The previous sections of this book have concentrated on using propositional or predicate calculus for describing the functions of a software system. For example, you have now been given the mathematical tools to express the essence of the paragraph lf all the monitoring valves are functioning and at least one computer is on-line, then, provided the system is in a functioning state, the header line will be displayed on all the VDUs attached to the system.
Functions and sequences
Authors: Ince D C
Published: November 2023

To outline the nature of functions. To describe a particular type of function known as a sequence. To describe two small case studies, one using functions, the other using sequences.7 .1 Functions A function is an important type of relation. It has the property that each element of its domain is associated with just one element of its range.
The specification language Z
Authors: Ince D C
Published: November 2023

To describe the schema, the main structuring device in the Z specification language. To describe how schemas can be included within each other. To outline how software systems can be described in the Z specification language. Although in the previous chapters computer systems have been described in precise terms, there has been little stress on the syntax of the mathematical notations used. This part of the book remedies this state of affairs. It describes a language called Z which is based on typed set theory and which allows a concise expression of the functional properties of computer systems. One of the principal features of Z is an object known as a schema.
Some small Z examples
Authors: Ince D C
Published: November 2023

To describe some small but relatively realistic examples of Z specifications. This section describes three examples of Z specifications. Each specification describes a part of a relatively realistic software system, with some degree of simplification introduced in order to keep the chapter to a manageable size. One of the more important parts of a computer’s operating system is the subsystem which maintains files which have been created by users. Part of the filing subsystem is the block handler. Files in the file store will be made up of blocks of storage which are held on some file storage device. During the operation of the computer, files will be created and deleted. This means that, in the first case, blocks will be needed to form a file and, in the second case, blocks will be released from a file ready for use in another file. In order to cope with this the filing subsystem will maintain a reservoir of unused blocks, and will also keep track of blocks which are currently in use. When blocks are released from a deleted file they are normally added to a queue of blocks waiting to be added to the reservoir of unused blocks. This is shown in Figure 11.1.
A large Z specification
Authors: Ince D C
Published: November 2023

To revise many of the concepts described in the previous three chapters. To show how a large Z specification might be structured. The purpose of this chapter is to demonstrate some of the power of the schema calculus in incrementally describing a system and to describe how error processing can be accommodated in a Z specification. The final part of the chapter describes one way of organizing a Z specification and outlines how natural language can be used in conjunction with schemas. The specification described in the chapter is for an automated system for a university library. Only selected parts of the statement of requirements are shown; for example, parts 2.8 to 2.11 dealing with the requirements for microfiche hard copy are omitted.
Lowenheim–Skolem theorems and their consequences
Authors: Manzano María
Published: November 2023

In 1915 Lowenheim proved that if a formula is satisfiable, then it has a countable model. The various statements of the Lowenheim–Skolem theorems which appear in this chapter are successive reinforcements of this initial theorem. Many of these theorems were proved even before model theory had obtained citizenship, and by methods which were very different from those which became commonplace afterwards. The proofs that I give are, with slight modifications, those from Tarski and Vaught’s (1957).
Abductive reasoning in three-valued logic for knowledge bases
Authors: Mathieu Philippe , Delahaye Jean-Paul
Published: November 2023

In this chapter we study the case of monotonic rule-based systems which use a forward chaining algorithm to make deductions and a query algorithm to compute queries to be asked of the user in order to draw closer to the expected goal (as Guru [11], IS [12] or Nexpert [18]). Unfortunately, in many systems the query algorithm is not consistent with the deduction algorithm. Queries asked by these systems are often useless. We show that computing a query for these systems is equivalent to abductive reasoning in three-valued logic and we propose a formalization of useful queries for expert systems. We give two definitions of intelligent queries and two levels on which to compute them: the logical level which aims at computing a list of intelligent queries, and the heuristic level which aims at choosing which query on that list it will ask of the user.
Updates and subjunctive queries
Authors: Grahne Gösta , Mendelzon Alberto 0.
Published: November 2023

A subjunctive query of the form ϕ > Ψ means ‘if ϕ were true in the knowledgebase, would Ψ also necessarily be true?’ We propose the following semantics for subjunctive queries: ϕ > Ψ will hold in the current knowledgebase T if Ψ holds in the result of updating T with ϕ . This is known as the Ramsey test in philosophy. We adapt the model checking approach of Halpern and Vardi: a knowledgebase is a finite set of finite sets of positive facts interpreted in a closed world setting. We then use Winslett’s possible models approach to give semantics to knowledgebase updates, and we introduce a query language which is essentially propositional logic, augmented with a subjunctive conditional that has an intensional interpretation in our model. We show that query answering and update can be performed in polynomial time in the size of the knowledgebase. However, query equivalence is shown to be complete in polynomial space, and this is also the complexity of query answering as a function of query size. We give a sound axiomatization of query equivalence and show that the update operator satisfies the postulates for updates adapted by Katsuno and Mendelzon from the Alchourr6n-Gardenfors-Makinson belief revision postulates.
Interior point algorithms for network flow problems
Authors: Resende Mauricio G C , Pardalos Panos M
Published: November 2023

A large number of problems in transportation, communications and manufacturing can be modelled as network flow problems. In these problems one seeks to find the most efficient, or optimal, way to move flow (e.g. materials, information, buses, electrical currents) on a network (e.g. postal network, computer network, transportation grid, power grid). Many of these optimization problems are special classes of linear programming problems, with combinatorial properties that allow the development of efficient solution techniques. In this chapter, we limit our discussion to linear network flow problems. For a treatment of non-linear network flow problems, the reader is referred to [17, 28, 29, 48].
What is a Deductive System?
Authors: Lambek Joachim
Published: November 2023

We discuss various styles of presnting deductive systems, with and without structural rules, paying some attention to equality between deductions. A deductive system should deal with (a) formulas, (b) deductions, alias proofs, (c) equality between deductions.
What is a Logical System?
Authors: Gabbay D M
Published: November 2023

There is an increasing demand from computer science, linguistics and philosophy for a variety of logical systems. This is prompted by the extensive applications of logic in theoretical computer science, artificial intelligence and logic programming. In these fields there is a growing need for a diversity of semantically meaningful and algorithmically presented logical systems which can serve various applications. Therefore renewed research activity is being devoted to analysing and tinkering with old and new logics.
Introduction
Authors: Russ Steve
Published: November 2023

Bernard Bolzano was a Christian humanist who devoted a lifetime of thought and writing to a far-reaching and wide-ranging reform of the representation, organization, and discovery of knowledge. He was ordained as a Catholic priest in 1805 and thereafter, for almost fifteen years, he held a university post in Prague lecturing in theology and giving regular ‘edifying discourses’ to both the students and the public. He was eventually dismissed from his post by the Hapsburg Emperor, Franz I, because his public views, on social and political issues were deemed dangerously liberal, and his theological views even heretical.
Geometry and Foundations
Authors: Russ Steve
Published: November 2023

The first of Bolzano’s publications (BG), on geometry, has several passages treating foundational or philosophical matters. His next publication (BD) on the methodology of mathematics was explicitly directed at addressing particularly the problems of geometry; it seems natural therefore to group these two together. The later work DP includes, curiously, several groups of geometrical definitions that are not actually used in the work itself, but offer a striking proposal for the problem of giving satisfactory definitions of geometric extensions (such as line, surface, and solid). A common theme connecting these three works is what Johnson calls Bolzano’s ‘definitional problem’ (the problem is that of defining the geometric extensions just mentioned, see Johnson (1977)).
Construction Consultations
Authors: Kangshen Shen , Crossley John N , Lun Anthony W -C
Published: November 2023

The 28 Problems of this Chapter are principally concerned with volumes of earthworks, capacity of barns of different shapes and work done by labourers. They may be classified into three groups:
A new statistical approach to granular mixtures
Authors: Mehta Anita , Edwards S F
Published: November 2023

There has been great interest recently in the theoretical (Edwards 1988; de Gennes 1988; Mehta and Edwards 1989) and experimental (Douady et al. 1989; Evesque and Rajchenbach 1989; Jaeger et al. 1989) properties of granular materials. This has concerned itself with the statics-how such materials pack, what configurations are observed as a function of their typical parameters-as well as the dynamics-the statistics of the avalanches observed when a sandpile is perturbed, the nature of the instabilities observed as a function of the frequency and amplitude of the perturbation-of such systems.
Spatial and spectral properties of fractons in percolating structures
Authors: Burston R A , Hughes, G D , Lambert C J
Published: November 2023

During the past 10 years there have been many papers aimed at elucidating the properties of random percolating structures. Many of these have dealt with static exponents characterizing properties such as elasticity, conductivity, or mass density. More recently, attention has shifted towards dynamical or quantum properties, such as the density of states per unit volume (DOS) or degree of localization of fracton modes.
The S = 1 quantum spin chain: some exact and numerical results
Authors: Paczek D , Parkinson J B
Published: November 2023

There has been great interest recently in quantum spin chains with S = 1. Much of this work was stimulated by Haldane’s (1983a, b) discussion of the isotropic Heisenberg model for which he predicted the existence of a disordered ground state (i.e. one in which the correlations decay exponentially) and a gap to the first excited state at k = n.

We consider a system with isotropic Heisenberg and biquadratic exchange, and a magnetic field, described by the Hamiltonian.
Single- and two-particle interference
Authors: Loudon Rodney
Published: November 2023

The reconciliation of wave-like optical interference phenomena with the particle-like properties of the quantized radiation field has intrigued physicists since the early years of the present century. Two main topics of discussion have been the effect of very low light levels-of the order of one photon at a time in the interferometer-on the interference fringes, and the possibility of determining which of the two optical paths through the interferometer were taken by the photons that produce the fringes. The first experiment with very low light levels was performed by Taylor (1909) at the instigation of J. J. Thomson. He found that Young’s interference fringes persisted when the low light levels were compensated by long photographic exposure times.
Trends within hydrogenic species: an empirical approach
Authors: Goltzene A , Schwab C
Published: November 2023

In contrast to the wealth of theoretical approaches developed for the computation of energy levels, the ab initio calculations of the electronic structures of defects in a crystal have lagged behind. This situation might be surprising for the experimentalist as a number of spectroscopic data are usually less exploited than they could be. For instance, in electron paramagnetic resonance (EPR), once the features of the spectra related to the number of components and symmetry have been taken into account, the values of the splitting parameters or Lande factors usually remain ignored. This has, however, a practical consequence as it leads to an inability to deal properly with the problem in the case of unresolved hyperfine spectra even if met in promising compound semiconductors such as the 111-V materials.
Omar Khayyam
Authors: Coolidge Julian Lowell , Gray Jeremy
Published: November 2023

Ghizathudin Abulfath Ibn Ibrahim Al Khayyami, who is generally known under the name given at the beginning of this chapter, spent most of his life at Naishapur. There is a good deal of uncertainty about his dates, but he seems to have lived during the latter part of the eleventh and the beginning of the twelfth centuries. There is a tradition that he died in 1123, but we cannot place too much confidence in it. The most significant episode in his early years was his friendship with two unusual young men, Nizam al Mulk and Hassan ibn Subbuh.
Antoine Arnauld
Authors: Coolidge Julian Lowell , Gray Jeremy
Published: November 2023

It is really noteworthy that the small fraternity of the Port Royal produced two writers of uncommon mentality who had more than a passing interest in mathematics. One of these was Blaise Pascal, whom we discussed in the last chapter, and who stood very close to the company without being technically a member of it, the other was his intimate associate Antoine Arnauld, ‘le grand Arnauld’ as his countrymen frequently called him. His intimacy with Pascal was so close that n some places it is difficult to distinguish between them, a difficulty which has arisen more than once in the long history of mathematics. Pascal had original, brilliantly original, ideas, and made important contributions towards the advancement of mathematical science.
Introduction
Authors: Gillies Donald
Published: November 2023

In modern English, ‘revolution’ is most widely used in a political context, and talk of revolutions in science or mathematics can, with justification, be considered as the application of a political metaphor to the development of science or mathematics. H will help us to evaluate this metaphor if we begin with a brief consideration of some political revolutions.
T. S. Kuhn’s theories and mathematics: a discussion paper on the ‘new historiography’ of mathematics (1976)
Authors: Mehrtens Herbert
Published: November 2023

In a paper published recently in Historia Mathematica, M. J. Crowe tried ‘to stimulate discussion of the historiography of mathematics by asserting ten “laws” concerning change in mathematics’ (Crowe 1975, p. 16). His starting point is the ‘new historiography of science’, whose basic book is T. S. Kuhn’s essay The structure of scientific revolutions (1970a), the. first edition of which appeared in 1962. There has been much discussion on Kuhn’s theses since. Eventually Kuhn had to refine some of his concepts, which was done in the important ‘Postscript 1969’ to the second edition of his book, and in further papers (Kuhn 1970b, 1970c).
Application of dynamic light scattering to biological systems
Authors: Janmey Paul A
Published: November 2023

In many respects, biological macromolecules are well suited for study by dynamic light scattering. They are usually sufficiently large to be strong scatterers at low concentrations, and their diffusion constants are generally appropriate to give rise to intensity autocorrelations that can accurately be measured by available instrumentation. Most important, biological macro-molecules are often synthesized in precisely defined molecular weights, allowing almost perfectly monodisperse solutions. These features have motivated many studies of the conformation and size of proteins, nucleic acids, polysaccharides, lipids, and their supermolecular assemblies. The sizes of biological molecules and assemblies that have been studied by DLS range from monomeric proteins of molecular weight as low as 6000 to intact cells several micrometres in diameter. A common experimental limit for these molecules is imposed by their thermal instability and propensity to aggregate or otherwise change their state of assembly in non-physiological solutions. The non-perturbing nature of DLS and the lack of a requirement for labelling the molecule of interest have facilitated measurements of very delicate materials and have allowed the detection of interactions between chemically distinct materials such as protein-lipid interactions under conditions designed to mimic those encountered in vivo. Interpretation of scattering from the wide variety of biological materials presents a formidable challenge to theorists and in some cases limits the structural information that can be derived from chemically well-defined samples. Therefore DLS measurements are often combined with such methods as electron microscopy, birefringence, static light scattering, or neutron scattering. to achieve an unambiguous molecular interpretation.
Elements Ii: The Dimension Of Squares
Authors: Fowler D H
Published: November 2023

These detailed mathematical arguments underly a historical argument: I propose that this whole programme may indeed correspond to an early Greek investigation of which only the basic geometrical figures survive. Thus the situation is complementary to that which we find in the Meno: our surviving manuscripts of Plato’s dialogues do not-and perhaps none of them ever did-contain any of the figures on which the mathematical arguments depend.
Babylon
Authors: Macintosh Wilson Alistair
Published: November 2023

Mesopotamia, the ‘land between the rivers’, is the name given by the Greek historian Polybius (second century BC) to the region enclosed between the Tigris and the Euphrates in southern Iraq (Fig. 4.1). In the 400 miles between Baghdad and the Persian Gulf, the land drops only about 30 feet (9 metres). The great rivers run slowly, often changing their courses and depositing enormous quantities of fertile silt. It has been calculated that the Tigris and Euphrates remove three million tons of eroded material a day from the highlands of Iraq, and deposit it on the plains. The resulting sediments carpet the southern part of Iraq with fertile soil, which ranges in depth from 16 to 23 feet (5 to 7 metres).
The Achaeans
Authors: Macintosh Wilson Alistair
Published: November 2023

About 2000 BC, a branch of the Indo-European or Aryan people was forced southward into what is now Greece. These invaders were later called the Achaeans. Their greatest poet, Homer, called them the Danaans or the Argives. They were to give the world the modern form of mathematics. The Achaeans were probably fairer skinned than the tribes living along the side of the Aegean sea, whom they conquered and merged with. Like the Shang, they fought on foot and from horseback, their greatest pleasures being fighting and hunting. They were great breeders and trainers of horses. When they finally settled, the Achaeans lived in houses which consisted of long rectangular rooms, with side walls projecting to form a shallow porch.