bluuebunny/crossref_metadata_2025_split huggingface dataset:
Integral Equations and Picard’s Method
Authors: Collins Peter J 
Published: November 2023

Abstract
Four main types of integral equations will appear in this book: their names occur in the table below. Suppose that f : [a, b] → ℝ and K : [a, b]2 → ℝ are continuous, and that λ, a, b are constants. We will search for continuous solutions y = y(x) to such equations. On occasion, x may range over a different domain from [a, b]; in which case, the domains of f and K will need appropriate modication. The function K = K(x, t) appearing in all four equations is called the kernel of the integral equation. Such a kernel is symmetric if K(x, t) = K(t, x), for all x, t ε [a, b]. A value of the constant λ, for which the homogeneous Fredholm equation has a solution y = y(x) which is not identically zero on [a, b], is called an eigenvalue, or characteristic value, of that equation, and such a non-zero solution y = y(x) is called an eigenfunction, or characteristic function, ‘corresponding to the eigenvalue λ’. The analogy with linear algebra is not accidental, as will be apparent in later chapters.
Existence and Uniqueness
Authors: Collins Peter J 
Published: November 2023

Abstract
We begin this chapter by asking the reader to review Picard’s method introduced in Chapter 1, with particular reference to its application to Volterra equations. This done, we may fairly speedily reach the essential core of the theory of ordinary differential equations: existence and uniqueness theorems. To see that this work is essential, we need go no further than consider some very simple problems. For example, the problem of finding a differentiable function y = y(x) satisfying has the solution y = (1 – x)–1 which does not exist at x = 1; in fact, it tends to infinity (‘blows up’) as x tends to 1. Further, most differential equations cannot be solved by performing a sequence of integrations, involving only ‘elementary functions’: polynomials, rational functions, trigonometric functions, exponentials and logarithms. The celebrated equation of Riccati, is a case in point, amongst the most simple examples. In Exercise 2, the reader is asked to show that the method of proof of our main theorem provides a practical method of seeking a solution of this equation. In general, the theorem provides information about existence and uniqueness without the need for any attempt at integration whatsoever.
Hilbert–Schmidt Theory
Authors: Collins Peter J 
Published: November 2023

Abstract
In this chapter, we shall recount some elements of the Hilbert–Schmidt theory of the homogeneous Fredholm equation where λ is a constant and the kernel K = K(x, t) is real-valued, continuous, and symmetric: We recall that an eigenvalue of (H) is a value of λ for which there is a continuous solution y(x) of (H), which is not identically zero on [a, b]. Such a y(x) is an eigenfunction corresponding to the eigenvalue λ. Notice that all eigenvalues must necessarily be non-zero. Although the material in this chapter fits nicely into our discussion of integral equations in Chapters 8–10, the pay-off comes in Chapter 12, as it is the widely applicable Sturm–Liouville equation that provides an important context where symmetric kernels naturally arise. For the next section only, we shall consider the case where eigenfunctions can be complex-valued. The reader will want to note how the theory for real-valued functions fits into this general case.
Project 2
Authors: Grandine Thomas A 
Published: November 2023

Abstract
Solve has just been asked to help out the Computer Operations Network For Useful Synthesis of Engineering Devices, a division of ACME which builds small electronic devices. CONFUSED has a new project currently in the mill: they are building the new ACME Widget For Useful Logarithms, a simple scientific calculator. You have been asked to help out the project team designing the AWFUL.
Project 14
Authors: Grandine Thomas A 
Published: November 2023

Abstract
The Medical And Legal Professions’ Research Associates for Computerized Technical and Industrial Complex Enzymes, a division of ACME which develops and manufactures medical products, has just come out with a new wonder drug, cramerrulic, used for treating cancer patients. It does a terrific job of controlling the growth of many kinds of malignant tumors, but it has a very undesirable side effect: even in moderate doses, it has a tendency to kill the patient. The drug has not yet been approved by the FDA, but MALPRACTICE has received permission to prescribe it to patients whose chances of survival are almost non-existent without it.
Project 16
Authors: Grandine Thomas A 
Published: November 2023

Abstract
ACME is currently under contract to help the Corporation Of Nuclear Electrical Devices, one of the nation’s largest commercial producers of electricity, build a laser fusion reactor. This reactor will be powered by a single strong laser pulse targeted at some small object in the middle of this device. The details of how this works are unimportant, but the characteristics of the laser pulse, at least for this particular project, are crucial. The people in CON ED’s laboratory have made several measurements of the laser pulse during the time when it is turned on, but they still have no idea what the pulse looks like as a function of time. Because of your excellent reputation in doing numerical computations, the people at CON ED have decided that they would like to have you produce a picture of this function. Your boss has agreed that you are just the person for the job.
Groups
Authors: McLarty Colin 
Published: November 2023

Abstract
This chapter treats groups as an example of structures in a category. It presumes some familiarity with groups, and is not a prerequisite to later chapters. We begin with a category C with all finite products, called the base category. For convenience we suppose that C has a selected product A × B, p 1, p 2 for each pair of objects A and B. If C is in fact Set the definition gives the usual groups. A group in Data is a data type together with computable multiplication and inverse rules, a group in Top is a continuous group, and in Man it is a Lie group.
The internal language
Authors: McLarty Colin 
Published: November 2023

Abstract
For more complex reasoning in a topos E, we introduce a notation combining variables with E arrows to make up a kind of set-theoretic language with a familiar (although not quite classical) logic. This language is called the internal language of E. The internal language, and how its terms refer to E arrows, is described in Section 14.1. The logic of this language is described in Section 14.2, and various proofs in it are given in Section 14.3.
Topologies
Authors: McLarty Colin 
Published: November 2023

Abstract
In this chapter is described Lawvere and Tierney’s elementary treatment of the topologies on toposes that Grothendieck first described. We use any base topos E and, given a topology j on E we construct a topos Ej which is ‘E modified by making j-true formulas actually true’. Grothendieck toposes are described.
The effective topos
Authors: McLarty Colin 
Published: November 2023

Abstract
In the effective topos, Eff, described by Hyland using suggestions from Dana Scott, Church’s thesis is true in the strong sense that every function from the naturals to themselves in recursive. In fact, all functions between objects built from the natural numbers, including the rationals and the real numbers, are recursive. Thus those objects are naturally suited to modeling data types in programming languages, and Elf is a natural setting for higher order recursion theory.
An Introductory Courses in Commutative Algebra
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
This book aims to be a concise introduction to topics in commutative algebra, with an emphasis on worked examples and applications. It combines elegant algebraic theory with applications to number theory, problems in classical Greek geometry, and the theory of finite fields which has important uses in other branches of science. Topics covered include rings and Euclidean rings, the four-squares theorem, fields and field extensions, finite cyclic groups, and finite fields. The material covered in this book prepares the way for the further study of abstract algebra, but it could also form the basis of an entire course.
Sketches of an Elephant A Topos Theory Cornpendiurn
Authors: Johnstone Peter T 
Published: November 2023

Abstract
Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and thereby to demonstrate the overall unity of the subject. The material is organized in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text.
Numerical Methods for Dynamical Systems
Authors: Beyn Wolf-J iirgen 
Published: November 2023

Abstract
These lectures are intended to give a survey of numerical methods for analyzing dynamical systems. The growing interest in these systems, in particular in their chaotic behaviour, has stimulated an immense amount of theoretical as well as numerical investigation. Therefore we restrict ourselves to a few typical phenomena which are only first steps to chaos, but whose numerical properties are fairly well understood. Among the topics covered are the numerical computation of invariant sets such as stationary points, periodic orbits and tori, the transitions between these objects in parametrized systems and the analysis of the longtime behaviour of numerical trajectories which are generated with sufficiently small step size. In particular, we discuss in some detail the numerical computation of singular points and of homoclinic orbits via defining equations.
Mathematical preliminaries
Authors: Howie John M 
Published: November 2023

Abstract
Most of the mathematical equipment we shall need is quite elementary, and many readers will be able to skip lightly over this chapter. The algebraic ideas in Sections 1.3 to 1.7 are required primarily in Chapters 3 and 7, and some further bits of algebra, required only in Chapter 7, are deferred until Section 7.1.
Algebraic Computing with Reduce
Authors: MacCallum Malcolm A H , Wright Francis J 
Published: November 2023

Abstract
REDUCE is one of the most widely available and simple to use computer algebra systems. It enables users to manipulate complex algebraic expressions and equations symbolically just as mathematicians and scientists do traditionally on paper, and its capabilities include differentiation and exact symbolic integration. This book provides a comprehensive introduction to REDUCE from starting to run it, through to using some of the user-contributed REDUCE packages written for specific applications. The authors' aim is to enable all students using REDUCE for the first time to gain a familiarity with the full range of REDUCE commands whilst at the same time learning something of the internal workings of REDUCE. Throughout, numerous exercises are provided to illustrate themes covered in the text as well as to encourage `hands on' working with REDUCE. Both authors are expert users of REDUCE and the text is based on their many years of teaching the system to undergraduate and graduate students. As a result, all those coming to use REDUCE for the first time will find this an invaluable tutor and guide. For more advanced users, it covers a number of aspects not included in the REDUCE manual.
Factorization and integration in Reduce
Authors: MacCallum Malcolm A H , Wright Francis J 
Published: November 2023

Abstract
The purpose of this chapter is to give an introduction to the algorithms employed by two of the most important of Reduce’s algebraic packages, namely the factorizer and integrator, without, however, going in full detail into the underlying pure mathematics. We do not give details of the big float package, although it is one of the largest pieces of Reduce source code, nor of the several smaller packages such as the internal editor, matrix handler and so on.
The user-contributed Reduce packages
Authors: MacCallum Malcolm A H , Wright Francis J 
Published: November 2023

Abstract
The capabilities of the basic Reduce system are being progressively extended by a library of “packages”, written mostly in symbolic mode by “users” rather than those primarily developing Reduce. These packages currently fall into two classes: those that are distributed with the system (since version 3.3), and those that are not distributed but are available to Reduce users. The former have mostly existed for longer and been more widely tested than the latter, and have a slightly more official status.
Groups of Finite Morley Rank
Authors: Borovik Alexandre , Nesin Ali 
Published: November 2023

Abstract
The book is devoted to the theory of groups of finite Morley rank. These groups arise in model theory and generalize the concept of algebraic groups over algebraically closed fields. The book contains almost all the known results in the subject. Trying to attract pure group theorists in the subject and to prepare the graduate student to start the research in the area, the authors adopted an algebraic and self evident point of view rather than a model theoretic one, and developed the theory from scratch. All the necessary model theoretical and group theoretical notions are explained in length. The book is full of exercises and examples and one of its chapters contains a discussion of open problems and a program for further research.
A second variety of fixed point theorems
Authors: Smullyan Raymond M 
Published: November 2023

Abstract
There is a result in combinatory logic known as the “second fixed point theorem,” whose statement, proof, and applications to undecidability will be dealt with in Part II of this chapter. It is based on methods of naming combinators within combinatory logic itself, and this fits in ideally with the overall purpose of this volume. Part I of this chapter is devoted to adequate naming functions. The second fixed point theorem itself has a host of variants, all of which have pleasant generalizations to sequential systems, which will be dealt with in our final chapter.
Some classical fixed point arguments compared
Authors: Smullyan Raymond M 
Published: November 2023

Abstract
In the last chapter we saw how fixed points are related to self-reference. We shall now look at fixed points in a larger perspective. Although no prior knowledge of mathematical logic, recursion theory, or combinatory logic is presupposed, Part I of this chapter consists of abstract versions of classical results in these three areas. We present them as problems, which the reader might have fun trying to solve. Solutions are given at the end of Part I.
Introduction to formal systems and recursion
Authors: Smullyan Raymond M 
Published: November 2023

Abstract
This and the next two chapters are written primarily, but not exclusively, for those readers with no prior knowledge of recursive function theory. We develop those elementary portions of the theory that will be needed for the remaining chapters. Our approach to the theory is via elementary formal systems, as defined in T.F.S. (Theory of Formal Systems). These chapters might also be of interest to readers who are familiar with recursion theory, but not with elementary formal systems, since these systems will be seen to facilitate a neat and simple proof of Gödel’s incompleteness theorem for first-order arithmetic. Also, readers who are familiar with T.F.S. might find Chapter 8 of interest in that it contains an improved method of arithmetizing metamathematics.
A garden of G-spaces
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
This chapter is intended to exhibit a variety of group actions that illustrate the concepts introduced in Chapter 3, and to sow some seeds that will later provide food for thought. Although we limit ourselves here to little more than sketches, each of the examples is of some importance in some branch of mathematics, and will be treated in greater detail in later chapters.
Complex numbers and quaternions
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
Complex numbers first appeared in the early 17th century as a way of expressing the •impossible’ roots of an algebraic equation with real coefficients with the help of the ‘imaginary’ square root i of - 1. The representation of a complex number a+ ib by the point of the Euclidean plane with coordinates (a, b) in a rectangular cartesian coordinate system. known to us as the Argand diagram, appeared at the end of the 18th century, and was of vital importance to the development of the theory of functions of a complex variable which took place in the 19th century. Although the word ‘imaginary’ persists in the terminology of’real part’ and ‘imaginary part’, complex numbers ceased to be mysterious, and as more confidence was gained in handling them it was possible not only to use the Argand representation in reverse, as a way of assigning a complex coordinate to each point of the Euclidean plane, but also to think of extending the notion of complex numbers to ‘hyper•complex’ numbers which would serve a similar purpose in Euclidean space of higher dimension. It is these developments which concern us here.
Time and Change for Al
Authors: Galton Antony 
Published: November 2023

Abstract
Outside physics and philosophy one seldom reasons much about time and change in the abstract, but on the other hand reasoning about particular times and changes—dates, processes, events, and so on—is all-pervasive both in the specialist sciences and in everyday life. No account of reasoning can properly be called complete if it does not say something about how we reason about change; and if, as in Al, the aim is to simulate or duplicate human reasoning computationally, then the account of change required must be both extensive and detailed. In this section we survey six areas pertaining to Al in which a need for temporal reasoning has been felt, introducing some of the key issues arising in these areas; in later sections we examine these issues in greater detail.
Area, Lattice Points, and Exponential Sums
Authors: Huxley M N 
Published: November 2023

Abstract
In analytic number theory a large number of problems can be "reduced" to problems involving the estimation of exponential sums in one or several variables. This book is a thorough treatment of the developments arising from the method developed by Bombieri and Iwaniec in 1986 for estimating the Riemann zeta function on the line s = 1/2. Huxley and his coworkers (mostly Huxley) have taken this method and vastly extended and improved it. The powerful techniques presented here go considerably beyond older methods for estimating exponential sums such as van de Corput's method. The potential for the method is far from being exhausted, and there is considerable motivation for other researchers to try to master this subject. However, anyone currently trying to learn all of this material has the formidable task of wading through numerous papers in the literature. This book simplifies that task by presenting all of the relevant literature and a good part of the background in one package. The audience for the book will be mathematics graduate students and faculties with a research interest in analytic theory; more specifically, those with an interest in exponential sum methods. The book is self-contained; any graduate student with a one semester course in analytic number theory should have a more than sufficient background.
Analytic lemmas
Authors: Huxley M N 
Published: November 2023

Abstract
We call the function f(x) the exponent (function), and g(x) the weight (function). A pure exponential sum or integral is one in which the weight function is unity. The first lemma enables us to insert a weight function g(x) whenever we have a uniform upper bound for a pure sum or integral. The other lemmas in this section give upper bounds for exponential integrals.
The Peirce Decomposition In A Jordan Algebra
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
In this chapter we introduce the Peirce decomposition first with respect to a single idempotent, then with respect to a complete system of orthogonal primitive idempotents, or Jordan frame. Much of this could be done more generally, but we restrict ourselves throughout to the case of Euclidean Jordan algebras, for which we have simple direct proofs in the spirit of our proof of the Spectral Theorem given in Chapter III. A basic result (Theorem IV.2.1) shows that one can make computations in a general Euclidean Jordan algebra in a manner rather similar to the special case of real symmetric matrices. Another important result (Corollary IV.2.7) states that, in the irreducible case, the group K acts transitively on the set of all Jordan frames. In section 3 we discuss the boundary structure of symmetric cones, this turns out to be closely related to the algebra developed in the first two sections. In the last section we give a brief discussion of the representations of a Jordan algebra; the Peirce decomposition gives rise to an important example of a representation.
Classification Of Euclidean Jordan Algebras
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
In this chapter, for the sake of completeness, we will give a proof of the classification of simple Euclidean Jordan algebras. The case of rank r = 2 was settled in Corollary IV.1.5. When r ≥ 3, such an algebra turns out to be isomorphic to a space of Hermitian matrices with entries in one of the algebras ℝ, ℂ, ℍ, or 𝕆. These algebras are characterized by the existence of a multiplicative Euclidean norm, and the key to the classification will be the Peirce decomposition and Lemma IV.2.2 stating that, for x inVij, yin Vjk, with i,j,k distinct:
Fundamental equations of motion of fluids
Authors: Rahman Matiur 
Published: November 2023

Abstract
The study of fluid motion and other related processes can begin when the laws governing these processes have been expressed in mathematical form. Usually, in investigating fluid flow problems, the physical situation is described by a set of differential equations. Thus the solution of the differential equations will predict the fluid flow pattern. For a comprehensive derivation of these equations, the reader should turn to standard textbooks including Aris (1962). In this book, we shall avoid rigorous mathematical development of the equations.
Hall Polynomials
Authors: Macdonald I G 
Published: November 2023

Abstract
Let o be a (commutative) discrete valuation ring, p its maximal ideal, k — o/p the residue field. Later we shall require k to be a finite field, but for the present this restriction is unnecessary. We shall be concerned with finite o-modules M, that is to say, modules M which possess a finite composition series, or equivalently finitely-generated o-modules M such that p’M=Q for some r&gt;0. If k is finite, the finite o-modules are precisely those which have a finite number of elements.