bluuebunny/crossref_metadata_2025_split huggingface dataset:
Algebraic and Geometric Background
Authors: Goldman William m. 
Published: November 2023

Abstract
This chapter contains prerequisites for the main text. The previous chapter was meant to illustrate these ideas which are developed in greater generality here. After a fairly lengthy discussion of linear algebra in complex vector spaces (in particular real structures, Hermitian and symplectic structures), we review the analytic and differential-geometric background concerning Kähler manifolds. We emphasize symplectic and contact geometry. This chapter concludes with a discussion of “Heisenberg spaces” which model the boundary of complex hyperbolic space.
Wave Mechanics
Authors: Hannabuss Keith 
Published: November 2023

Abstract
According to Planck and Einstein the energy and frequency of light are related by E = ђw. De Broglie observed that to be consistent with the theory of relativity the momentum p of a wave should be ђk where k is the wave vector of magnitude 2π /wavelength, perpendicular to the wave front. (This is because the special theory of relativity combines E and p and also ω and k into four-dimensional vectors, and linear relations between particular components force the same relations for all other components as well. This is explained in more detail in Section 17.1, which is independent of the rest of the book.)
Quadratic and linear potentials
Authors: Hannabuss Keith 
Published: November 2023

Abstract
Although the classical examples of harmonic oscillators such as springs or the simple pendulum are of limited use in the submicroscopic world of quantum mechanics, there are plenty of other applications that make the oscillator worth studying. Atoms in a crystal, for instance, oscillate about their mean positions and it is their vibrations that carry sound and heat through the material. Indeed, we know from classical theory that by using normal coordinates we may regard any system performing small oscillations about a point of stable equilibrium as a collection of harmonic oscillators. This extends even to the normal modes of vibration of a string or of the electromagnetic field. The harmonic oscillator thus provides the key to understanding how wave phenomena such as light can be quantized, and lies at the heart of Einstein’s idea of the photon and of Planck’s quantum.
The hydrogen atom
Authors: Hannabuss Keith 
Published: November 2023

Abstract
Whilst theoretical physicists struggled to understand the interactions between matter and radiation, new experiments were revealing that atoms had internal structure. In 1897 J.J. Thomson showed that cathode rays consisted of a stream of negatively charged particles much lighter than any known atom. These became known as electrons. It was the 1896 discovery and subsequent investigation of radioactivity which forced the startling realization that, far from being indivisible and indestructible as had been supposed, atoms can disintegrate or be shattered into smaller pieces.
Symmetry in quantum theory
Authors: Hannabuss Keith 
Published: November 2023

Abstract
Physical systems often have some kind of obvious symmetry, such as the rotational symmetry of the central force field about a nucleus. Mathemati cally this means that the system is invariant under the action of a symmetry group G. (The main ideas of group theory are reviewed in Appendix Al.) Such a physical symmetry must also be present in quantum theory in a way that is compatible with the other important structures, namely the vector spaces and inner products. The simplest way in which this can happen is for G to act on the inner product space 1i by unitary transformations, since these respect both the linear structure and the inner product.
Iterative perturbation theory
Authors: Hannabuss Keith 
Published: November 2023

Abstract
The Rayleigh-Schrodinger theory described in the preceding chapter is by no means the only approach to stationary perturbation theory, nor is it usually the most powerful. As one of many possible alternatives we shall now describe an iterative procedure for approximating stationary states, sometimes called Wigner-Brillouin perturbation theory, which resembles the Feynman-Dyson method of Section 11.3.
Symmetries of elementary particles
Authors: Hannabuss Keith 
Published: November 2023

Abstract
It will have become apparent over the last few chapters that the electrons, protons, and neutrons, which are the most prominent constituents of mat ter, are only some of the many particles that are now known. There are also neutrinos, which appear in some nuclear reactions, Yukawa’s mesons, which help bind the nucleus together, and then to every particle a corresponding antiparticle. With improved techniques for detecting particles in cosmic rays and the increasingly powerful particle accelerators built after the war more and more different kinds of subatomic particle were discovered. The various conservation laws found to hold during collisions between particles suggested that there must be some kind of symmetry principle linking different particles. For example, protons and neutrons belong to a class of particles called baryons, and during collisions the total number of baryons minus the total number of antibaryons is constant. A similar conservation law held for the class of leptons which included the lighter particles such as electrons and neutrinos. Scattering experiments at higher energies began to indicate that the proton has an internal structure suggesting that it might be composed of still smaller particles.
Linear Matrix Equations
Authors: Lancaster Peter , Rodman Leiba 
Published: November 2023

Abstract
In this chapter we present a self-contained account of some well-known facts concerning solutions of linear matrix equations. Our attention is restricted mainly to the Lyapunov and Stein equations: SA-BS=Γ, and S-BSA=Γ, respectively (see equations (5.2.3) and (5.2.4) below). In particular, their symmetric forms (when B = A*, Γ * = Γ) are important in the sequel. Conditions are developed which ensure the (suitably defined) “stability” of solution matrices S of these equations.
Geometric Theory for the Discrete Algebraic Riccati Equation
Authors: Lancaster Peter , Rodman Leiba 
Published: November 2023

Abstract
It is to be shown in Chapter 16 that, to solve the “linear-quadratic regulator” (or LQR) problem for a time-invariant differential system, it is necessary to solve an algebraic Riccati equation of the kind discussed in Part II of this book, equation (7.2.1) for example. The discrete analogue of the LQR problem requires the solution of a rather different algebraic equation for X: It is therefore tempting to describe this equation as a discrete algebraic Riccati equation (or DARE) and, when convenient, we shall do so. The geometric theory for the DARE turns out to be considerably more complicated than for the symmetric CARE. The identification of a main operator (whose invariant subspaces determine solutions of a DARE) is already a significant task when compared to analysis of CARE (see Proposition 12.2.2). Having done this, it is possible to obtain several conclusions for the DARE by a suitable transformation between equations of CARE and DARE types (see Lemma 12.3.2). In contrast to analysis of the CARE it is convenient to make use of techniques involving rational matrix functions at an early stage of the analysis.
Linear–Quadratic Regulator Problems
Authors: Lancaster Peter , Rodman Leiba 
Published: November 2023

Abstract
The “linear-quadratic regulator” (LQR) problem of optimal control has probably provided the greatest single stimulus for investigation of matrix Riccati equations in differential, difference, and algebraic forms. In this chapter the continuous and discrete LQR problems are to be outlined and then the solutions of these problems using Riccati equations are developed. Consistent with the spirit of this book attention is confined to time-invariant (i.e., constant coefficient) problems, even though significant parts of the theory are readily extended to linear time varying problems (see, for example, Brockett (1970), Kalman, Falb and Arbib (1969), or Russell (1979)). In the first four sections the LQR problem for time-invariant, continuous systems is described. We do not use the line of argument initiated by Kalman which involves the technique of dynamic programming. In view of our limited objectives a more direct line of argument seems appropriate even though it is rather delicate. This seems to originate with Brockett (1970), and has been refined by Hijab (1987). Here, we adapt the Brockett-Hijab development to our needs. This line of argument also has the great advantage that it is readily adapted to the discrete LQR problem. This is done in Sections 16.5 and 16.6.
The Total Least Squares Technique
Authors: Lancaster Peter , Rodman Leiba 
Published: November 2023

Abstract
Notice that AXo = (AA+)B and, as AA+ is just the orthogonal projector onto Im A, Xo is the minimum norm solution of the consistent system AX = (AA+)B obtained by projecting Im B onto Im A. Thus, in this process, the subspace Im A plays the pivotal role and the “right-hand side” matrix B is “adjusted” to produce a solvable system. In many applications inconsistent systems arise in an attempt to model consistent equations which are corrupted by observation, modelling, or computational errors. The argument given above can be criticized in this context for holding The best least squares solution of linear systems remains a very important and widely useful technique after a 200-year history. The total least squares idea can be traced back for about 100 years, but has come into prominence more recently with investigations of numerical techniques
Customer requirements and specification
Authors: Ince D C 
Published: November 2023

Abstract
To examine the problems that occur with natural language based documents. To outline what makes a good specification. To detail a simple procedure for requirements analysis and system specification. To show how mathematics is a good medium for specification. The aim of this chapter is to examine some of the faults found in customer statements of requirements and software specifications written in natural language. The purpose in doing this is twofold. First, the document given to staff who produce the software specification is the customer statement of requirements. It is important that such staff realize the nature of such documents. Second, the inadequacies of natural language as a specification medium are most manifest in statements of requirements and software specifications. Before attempting to describe the use of mathematics in system specification, it will be useful to describe the faults that it should eliminate.
Lexical-domain object-feature logic
Authors: Sandewall Erik 
Published: November 2023

Abstract
We now proceed along the lines defined in Chapter 4, subsection 4.1.2 and introduce a lexical object domain alongside, but distinct from, the value domains for the features. This extension allows the logic to refer to object domains in the O component of scenario descriptions. I will refer to the resulting logic as lexical-domain object-feature logic, or LFL. Section 5.4 will outline how a terminological language can be defined on top of the LFL semantics, and therefore indicates a bridge to KL-ONE-type languages.
LFL does not contain a notion of time, so it can only be used for describing static or momentary phenomena. However, within LFL we define a concept of state which will be used extensively to repreSCDt the ‘snapshots’ or momentary states in later chapters.
Temporal feature logic for discrete time domains
Authors: Sandewall Erik 
Published: November 2023

Abstract
Earlier chapters have defined IDS developments whose structure is the five tuple 〈B, M, R, A, 〉. They have also defined the operation of chronicle completion where a scenario description is given, and where it is sufficient to first identify the set of developments that satisfy the given description, then to extract the set of weakened models of the form 〈M, R〉while throwing away the rest of the information, and then to use the set of weakened models as the set of intended conclusions from the given scenario description. In this way, one will only obtain conclusions about feature-values and changes of feature-values at various points in time, but not conclusions about, for example, the starting times and ending times of actions.
Intended models for chronicles in 𝒦-IA
Authors: Sandewall Erik 
Published: November 2023

Abstract
By the definitions in chapter 2, subsection 2.3.1, a chronicle in 𝒦-LA is a tuple 〈 𝒦, 0, 〈IA, LAW 〉, SCD, OBS 〉. Some of the previous chapters have defined the syntax of formulae which may occur in LAW, SCD, and OBS. Other chapters have defined an underlying semantics in terms of the ego-world game and the trajectory concepts. We are now ready to establish the connection between those two backgrounds, and to define the intended set of models for a given chronicle in terms of the underlying semantics. In this context we shall also introduce a number of auxiliary concepts, including syntactical constraints on the formulae in the premise partitions LAW, SCD, and OBS. Similarly we introduce some subsidiary model-theoretic concepts, such as the classical and intended models restricted to a finite time interval, which will be used in the subsequent proofs.
Model Theory
Authors: Manzano María 
Published: November 2023

Abstract
Logic languages are free from the ambiguities of natural languages, and are therefore specially suited for use in computing. Model theory is the branch of mathematical logic which concerns the relationship between mathematical structures and logic languages, and has become increasingly important in areas such as computing, philosophy and linguistics. As the reasoning process takes place at a very abstract level, model theory applies to a wide variety of structures. It is also possible to define new structures and classify existing ones by establishing links between them. These links can be very useful since they allow us to transfer our knowledge between related structures. This book provides a clear and readable introduction to the subject, and is suitable for both mathematicians and students from outside the subject. It includes some historically relevant information before each major topic is introduced, making it a useful reference for non-experts. The motivation of the subject is constantly explained, and proofs are also explained in detail.
Hypothetical reasoning with intuitionistic logic
Authors: Bonner Anthony J 
Published: November 2023

Abstract
This chapter addresses a limitation of most deductive database systems: they cannot reason hypothetically. Although they reason effectively about the world as it is, they are poor at tasks such as planning and design, where one must explore the consequences of hypothetical actions and possibilities. To address this limitation, this chapter presents a logic-programming language in which a user can create hypotheses and draw inferences from them. Two types of hypothetical operations are considered: the insertion of tuples into a database, and the creation of new constant symbols. These two operations are interesting, not only because they extend the capabilities of database systems, but also because they fit neatly into a well-established logical framework, namely intuitionistic logic. This chapter presents the proof theory for the logic, outlines its intuitionistic model theory, and summarizes results on its complexity and on its ability to express database queries. Our results establish a strong link between two previously unrelated, but well-developed areas: intuitionistic logic and computational complexity. This, in turn, leads to a strong link with classical second-order logic. Furthermore, unlike many expressibility results in the literature, our results do not depend on the artificial assumption that the data domain is linearly ordered.
Logic Without Model Theory
Authors: Kowalski Robert 
Published: November 2023

Abstract
Arguably, model theory serves two main functions: (1) to explain the relationship between language and experience, and (2) to specify the notion of logical consequence. In this chapter I shall propose the notion of ‘knowledge assimilation’, the assimilation of new information into a knowledge base, as an alternative understanding of the way in which a knowledge base formulated in logic relates to externally generated input sentences that describe experience. I shall argue that the notion of logical consequence can also be understood within a knowledge assimilation framework, in terms of sentences that must hold no matter what stream of input sentences might arise in the future.
The Transmission of Truth and the Transitivity of Deduction*
Authors: Tennant Neil 
Published: November 2023

Abstract
Frege assigned to logic the task of ‘discovering the laws of truth’ Logical proofs preserve truth from their premisses to their conclusions. Logic is also said to be a matter of form, not content. I intend in this essay to stay with considerations of just truth and logical form, in order to, address the question of the transitivity of deduction.
Logical Constants as Punctuation Marks*
Authors: Došxen Kosta 
Published: November 2023

Abstract
This chapter presents a proof-theoretical approach to the question ‘What is a logical constant?’ The approach starts with the assumption that logic is the science of formal deductions, and that basic formal deductions are structural deductions, i.e. deductions independent of any constant of the language to which the premises and conclusions belong. Logical constants, on which the remaining formal deductions are dependent, may be said to serve as ‘punctuation marks’ for some structural features of deductions. This punctuation function, exhibited in equivalences which amount to analyses of logical constants, is taken as a criterion for being a logical constant. The chapter presents an account of philosophical analysis which covers the proposed analyses of logical constants. Some related assumptions concerning logic are also considered. In particular, since a logical system is completely determined by its structural deductions, alternative logical systems arise by changing structural deductions while having constants with the same punctuation function. Some other approaches to the question ‘What is a logical constant?’, grammatical, model-theoretical, and proof- theoretical, are briefly considered.
Finitary Inductively Presented Logics*
Authors: Feferman Solomon 
Published: November 2023

Abstract
A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems) met in practice. A f.i.p. theory FS0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS0 is a conservative extension of PRA. The aims of this work are (i) conceptual, (ii) pedagogical and (iii) practical. The system FSo serves under (i) and (ii) as a theoretical framework for the formalization of metamathematics. The general approach may be used under (iii) for the computer implementation of logics. In all cases, the work aims to make the details manageable in a natural and direct way.
Field Measurement
Authors: Kangshen Shen , Crossley John N , Lun Anthony W -C 
Published: November 2023

Abstract
The Greek historian Herodotus (5th century BC) said that the civilization of Egypt was merely a gift of the Nile. Each year this river floods the land and leaves rich soil behind. Egyptian geometry is said by Herodotus to have originated in the need, created by this annual flooding, to redetermine the boundaries of the lands owned by the farmers.
Commensurable spin structures and their excitations
Authors: Mackintosh A R , Jensen J 
Published: November 2023

Abstract
The standard model of rare-earth magnetism was constructed during the 1950s and early 1960s, and Roger Elliott was one of its major architects. One of the first coherent descriptions of the theory is to be found in his classic discussion of the magnetic structures of the heavy rare-earth metals by means of the molecular-field approximation (Elliott 1961). With his colleagues, he also gave in the following year an account of the magnetic excitations (Cooper et al. 1962). In these, as in other early theoretical treatments of rare-earth magnetism, most of the results were obtained by analytical means, and the problems caused by the incommensurability of the lattice and magnetic periodicities were either ignored or treated by perturbation theory. In this chapter we shall adopt a mean-field method which is closely related to those used earlier, but differing from them in two important respects. There is convincing experimental evidence that the periodicity of magnetic structures such as the helix in the rare earths (and particularly in holmium which we shall be considering in some detail) tends to ‘lock in’ to some rational multiple of the lattice periodicity.
Helical spin systems: a progress report
Authors: Harris A B 
Published: November 2023

Abstract
The introduction of models to describe helical spin systems in rare-earth metals was initiated by the pioneering work of Roger Elliott (1961), to whom this volume is dedicated. The purpose of this chapter is to review some recent theoretical developments in this general area. We note that this topic of research has had continuing impetus from both experiment and theory. One family of experiments concerns the nature of magnetism in the metallic rare earths and this work is reviewed elsewhere in this volume. Also, we point out the existence of helical magnetic insulators of the chemical form TX2, where T represents a transition metal, such as Co or Ni and X is typically Br or I. The magnetic structure and some properties of such materials are well established (Adam et al. 1980; Day and Ziebeck 1980; Kuindersma et al. 1981). Furthermore, a recent experimental programme (Endoh et al. 1988), stimulated by the need to understand the high-temperature superconductors, had focused on the lanthanum cuprate materials, which display both quantum effects in the pure materials (Chakravarty et al. 1988, 1989) and helical spin correlations (Shirane et al. 1989) in the doped materials.
Dressed particles in the Hubbard chain
Authors: Kawakami Norio , Okiji Ayao 
Published: November 2023

Abstract
The Hubbard model has been studied for a long time as a simplified model for electron correlation problems such as the metal-insulator transition, itinerant magnetism, and so on. Moreover, the discovery of high 7;, superconductors has again raised an important question about the strong correlation problem near the metal-insulator transition, including low-dimensional quantum fluctuations (Anderson 1987). In recent works, we have precisely studied various thermodynamic properties of the one-dimensional Hubbard model (Kawakami et al. 1989; Usuki et al. 1989) and clarified the underlying physics near the Mott insulator in terms of the dressed particles of the Bethe ansatz method (Kawakami and Okiji 1989a, b). In this brief report, we present the exact calculation for the dressed particles in the Hubbard chain near the Mott insulator.
Pietro Dei Franceschi
Authors: Coolidge Julian Lowell , Gray Jeremy 
Published: November 2023

Abstract
The problem of representing three-dimensional objects accurately on a flat surface would seem naturally to be one that would attract the attention of both the mathematicians and the artists, and that at an early date. As a matter of fact both categories of persons left it severely alone until a surprisingly late time. The geometers may have thought it too simple to exercise their skill, the artists found it too mathematical to suit either their taste or their capacity.