bluuebunny/crossref_metadata_2025 huggingface dataset:
Collections
Authors: Potter Michael D
Published: November 2023

In every part of mathematics we study objects. Whether these objects be numbers, points in space, or elements of a locally convex topological vector space, they can be formed into collections. In this chapter (and indeed throughout the book) we shall study the properties which collections have simply by virtue of being collections, no matter what they are collections of In other words, we shall be concerned only with properties arising from the relationship which holds between an object and any collection to which it belongs: this relationship is called membership.
Relations
Authors: Potter Michael D
Published: November 2023

Now that we have an axiomatic base for the theory of collections in place we are in a position to define the familiar apparatus of pure mathematics: functions, equivalence relations, orderings, etc.
Basic set theory
Authors: Potter Michael D
Published: November 2023

None of the axioms we have stated so far prevents the possibility that all collections are finite. So if we want to be able to construct infinite collections in our theory-which we do-we must make the further assumption that such collections exist. There are two ways of doing this: we can assert either that there are infinitely many individuals (that, in the terminology suggested by Fig. 1, the universe is infinitely wide) or that there are infinitely many days (that the universe is infinitely long). We shall take the latter path here: we can achieve what we want by asserting the existence of at least one limit day. (The definition of ‘infinite’ will not be given until Chapter 4; in the meantime we rely on naive notions for our discussion.) With this addition we obtain a system called ZA which is adequate for the presentation of much of modern mathematics.
Cardinals
Authors: Potter Michael D
Published: November 2023

In this chapter we shall study in some detail the concept of ‘equinumerosity’, which was introduced in §2.2 to express in our theory the idea of collections having the same size. In order to deal with this notion it will be helpful to associate with each collection an object which we can look on as being its size.
Highest common factor
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

In this chapter we shall consider divisibility, common factors, and the existence of highest common factors in arbitrary integral domains. From this work we shall deduce properties of integers which, as well as being of interest in their own right, will be needed in later chapters. Although our main interest in the short term is the ring of integers, we shall prove some of the results in a more general form which will be useful for later applications. Roughly speaking, the sort of results which we are interested in for integers can usually be extended to arbitrary Euclidean domains with only minor modifications to the proofs.
Fields and polynomials
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

The two most important results which we shall prove in this chapter are the division algorithm for polynomials in one indeterminate over a field, and that the integers modulo p form a field for every prime number p (thus providing our first examples of finite fields). We gave the definition of a field earlier in Chapter 2. Fields and polynomials over them play an important role in algebra. In this chapter we collect together some of the elementary but important properties which we shall need later.
Homomorphisms, ideals, and factor rings
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

It is often the case that two rings for which we already have some information are in fact related to each other. For instance the ring 71./nl of integers mod n is in some sense derived from the ring 71. of integers. We shall now study such connections between rings. This is done by means of homomorphisms, which are the appropriate type of function from one ring to another. One reason for doing this is that it helps us to organize our examples of rings by establishing natural connections between some of them. Closely related to homomorphisms are ideals which are certain subsets of a ring. The properties of the two are intimately connected. This theory then allows us to construct new examples of rings with certain specified properties. We shall give an important application of this type in Chapters 14 and 15, where we shall show how to construct a finite field with pn elements for any choice of the prime number p and positive integer n.
Principal ideal domains and a method for constructing fields
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

The simplest way to construct an ideal I of a ring R is to take I to be the set of elements of R which are multiples of (or equivalently, are divisible by) some fixed element x ∈ R.
Finite fields
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

We shall show in this chapter that for each prime p and positive integer n there is one and only one field with pn elements. This field is sometimes called the Galois field of order pn and is denoted by GF(pn). We shall prove the existence of GF(pn) by showing that there is an irreducible polynomial of degree n over the field ‘11./pZ. (see 14.17). Finite fields have practical applications. The interested reader should consult Biggs (1985) for information in this area.
Geometric Morphisms - Basic Theory
Authors: Johnstone Peter T
Published: November 2023

In Chapters A1 and A2 we took the view that, whenever we introduced a new class of categories, there should be a corresponding class of functors, namely those which preserve the structure involved in the definition of the given class of categories. Thus the notion of topos, when we introduced it, came with the associated notion of logical functor. However, an important aspect of topos theory, and one which particularly reflects the confluence of geometrical and logical influences in the origins of the subject, is the fact that it has another natural notion of ‘morphism of toposes’ associated with it: the notion of geometric morphism. We shall see in 4.1.18 below (and again in B2.2.7) that geometric morphisms are also ‘structure-preserving functors’ in a suitable sense, although the structure they preserve is different from that involved in our original definition of a topos.
Internal And Locally Internal Categories
Authors: Johnstone Peter T
Published: November 2023

The theory of enriched categories is normally studied in the context of a base category V with a monoidal structure (usually, but not always, symmetric). However, we are primarily interested in studying categories enriched in a topos, where the monoidal structure is taken to be the cartesian one (i.e. that induced by finite products), and so we shall save ourselves a bit of extra work by developing the theory only for cartesian monoidal structures. Throughout this section, therefore, our base category V will be assumed to have finite products.
Toposes Over A Base
Authors: Johnstone Peter T
Published: November 2023

We are interested in this chapter in studying the 2-category Top/S of toposes over a fixed topos S, through the medium of indexed category theory. We recall the precise definition of this 2-category from Section A4.1: its objects (which we call toposes defined over S, or simply S-toposes) are geometric morphisms p: E — S with codomain S, its morphisms (q: F — S) — (p: E — S) are pairs (f, a) where f: F — E is a geometric morphism and a : q = pf a geometric transformation, and its 2-cells (f, a) — (g, fi) are geometric transformations f — g compatible in the obvious sense with a and fi. However, we shall almost invariably abuse notation by suppressing any mention of the 2-isomorphism a when specifying 1-cells of Top/S; we shall also tend to suppress the structural morphism p when specifying objects of Top/S, and simply write ‘E is an S-topos’. (The justification for the latter abuse is contained in Theorem 3.1.2 below.)
Rotating frames
Authors: Lunn Mary
Published: November 2023

When considering motion close to the surface of the Earth we have as yet no formalism which takes into account the rotation of the Earth about its north-south axis. When heavy artillery lire takes place over a range of 30 km or more the effect of the Earth’s rotation is detectable. In Chapter 1 we saw that the choice of fixed axes in an inertial frame S did not influence the way in which we used Newton’s laws, nor, more importantly, did a change to a new frame moving with uniform velocity with respect to S. However, if the new frame is specified by axes which are rotating with respect to S we have strong evidence to suggest the frame is not inertial. Most of us have visited the fairground. Some brave souls may even have paid for a ‘go’ on the ‘Wall’ in which the joy-riding clients stand with their backs to the interior wall of a large drum. The drum spins about its central axis. At maximum speed the floor of the drum is removed leaving the incumbents pinned to the wall. If the drum were not spinning everyone would fall down when the floor was removed. The natual frame of reference as seen by the clients in the drum is a frame fixed in that drum. If this frame were known to be an inertial frame, applying Newton’s second law in it would tell us that when the floor was removed the people would fall down the wall. The conclusion must be that a frame spinning with respect to an inertial frame is not itself inertial. Nevertheless it is often convenient to use a rotating frame of reference. The intention in the remainder of this chapter is to set up an equation derived from Newton’s second law in an inertial frame so that we are able to use rotating frames of reference.
Soluble problems in rigid body motion
Authors: Lunn Mary
Published: November 2023

A sphere is symmetrical under any rotation about its centre of mass. A cube does not have the total symmetry of a sphere. Yet if it is rotating about its centre of mass G its behaviour is very similar to that of the sphere, due to the simple form of the inertia tensor at G. In each case the inertia tensor is a multiple of the identity matrix.
Groups and geometries
Authors: Pillay Anand
Published: November 2023

If Dis aω-categorical, non-trivial strongly minimal set, we know from Chapter 2 that D is locally modular and its associated geometry is affine or projective geometry over a finite field F. Either way, as in section 6 of Chapter 2, we obtain a strictly minimal set D1 whose pregeometry is projective geometry over F. The well-known ‘coordinatization theorem’ (see Artin’s Geometric Algebra [A]) produces a definable infinite-dimensional vector space over F.
Least-Sq Uar Curve Fi S Spline Ting
Authors: Dierckx Paul
Published: November 2023

In this chapter we well show how the properties of B-splines can be fully exploited when calculating a least-squares spline with fixed knots. When freeing the position of these knots. the least-squares approximation criterion results in a difficult, non-linear and constrained optimization problem. We will explore a number of possibilities to deal with the constraints, present a specific optimization algorithm and propose a strategy to obtain a good initial position for the knots. Finally we will discuss a number of tests to determine the optimal number of knots, the ultimate key in the design of a completely automatic curve fitting algorithm.
More Scatt Red Data Smoothing
Authors: Dierckx Paul
Published: November 2023

A certain drawback of tensor product splines is that they are always defined on a rectangle. In applications with a well specified and non-rectangular approximation domain D their use may therefore be rather inconvenient. Of course, if D is bounded, we can always define a rectangle R containing D and then determine a spline approximation on R. However, this will not always give satisfactory results, especially if some additional constraints at the boundary of D must be fulfilled. Also, the inevitable fact that we will create (large) subregions of the rectangle without data points will increase the chance of rank deficient systems (see section 9,1.2) and may therefore endanger the numerical stability of the approximation algorithm. Another possibility is to search for a smooth transformation that maps the region D onto a rectangle R. Of course this will not always be possible. Besides, it may happen that we will have to impose additional constraints on the tensor product spline to guarantee that the corresponding function on D is still sufficiently smooth. As an illustration, we will show how the scattered data algorithims of Chapter 9 can be adapted for regions which can easily be described in polar coordinates. In particular, we will first consider the unit disk C and then, more generally, any region D that has a boundary which can be described by a smooth periodic function of the angular coordinate. As a second example we will consider the smoothing of data over the sphere. Here also additional boundary constraints will have to be imposed on the approximating tensor product splines, similar to those for the unit disk
Automata
Authors: Howie John M
Published: November 2023

The idea of an automaton has its origins in the scientific determinism of the seventeenth century enlightenment. Increasingly exact understanding of the physical world seemed to give the universe the characteristics of a machine, and as early as Descartes (15961650) there was speculation that the behaviour of animals, and even of man himself, could be described in mathematical terms. Arguments about determinism and freedom of the will in respect of human behaviour have raged for many centuries and will probably rage for many more. But as far as I know there has never been a claim that computers possess freedom of the will, and so while there may be some doubt as to the legitimacy of mathematical machine theories in describing the complex subtleties of human brain functions, there is general agreement that these theories help greatly not only in the design and operation of computers but also in understanding the limitations of a computational approach.
Languages
Authors: Howie John M
Published: November 2023

Up to this point we have freely used the word language’ for any subset whatever of A (where A is as usual our finite alphabet). Clearly this is for most purposes much too general a notion: any language spoken and understood by people or accepted by computers must have a degree of form and structure, and a proper theory of formal languages ought to mirror this fact. The key contribution in this area was made by Chomsky (1959), whose theory of formal grammars has had a major influence in the development of the subject. In this chapter we shall examine the so called ‘phrase structure grammars’ and then go on to study two special cases. Languages generated by ‘regular’ grammars turn out to be the familiar rational languages first encountered in Chapter 2 and hence are associated with finite state automata in the way that we have seen (Theorem 2.5.2).
Pushdown automata
Authors: Howie John M
Published: November 2023

Before introducing the main ideas of this chapter we shall find it useful to make a slight change in our informal ‘imagery’ concerning finite state automata. At a typical stage qj scans the leftmost square containing aj, deletes aj, changes to gj+1 and moves to scan the new leftmost square.
Turing machines
Authors: Howie John M
Published: November 2023

"In introducing the idea of a pushdown automaton in Chapter 5 we found it helpful to think of a word w = a1a2 an in A as printed on a tape with the initial state i scanning the leftmost square. An FSA can be seen as acting on w by constantly moving right, deleting a symbol at each stage. A PDA similarly can never move left, but it can stand still temporarily in order to make adjustments to its memory stack. It can print symbols in its memory stack, but can never add any new symbols to the tape. Part of the object of exploring a hierarchy of types of languages and machines is to examine the limits of computability. The FSA and the PDA can be seen as computing devices, and in the last few chapters we have explored their strengths and weaknesses. The next kind of machine to consider is the one devised by Alan Turing (1912-1954) and it is no accident that Turing was deeply involved in the development during the Second World War of the first computer, for with the Turing machine we reach the ‘;Rolls Royce’; among the machines described in this book and acquire the kind of power that corresponds to that of a real computer.
Reduce as an algebraic calculator
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

In this book we present almost all of the Reduce computer algebra system, plus some mathematical and computational topics that go beyond the basic system. Some technical remarks are included in an appendix.
Algebraic procedures, operators and algorithms
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

After looking at algebraic procedures and at further details of operators, we go on to use Reduce in some larger and more complete examples that involve some simple algebraic algorithms (for an introduction to algebraic algorithms in general see Davenport et al. 1988).
Basic Group Theory
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

This chapter contains a review of basic group theory and the group theoretic notation we use throughout. The exercises in this chapter are an integral part of the text and will be used freely in later chapters. Most readers will refer back to specific points as the need arises subsequently. More substantial background is developed in a more leisurely fashion in Chapter 3. The reader who agrees that the present chapter can be skipped for now may nonetheless wish to take note of the following points. Topics in Section 1.1 include some special aspects of the theory of nilpotent groups (Exercises –15), and a simple lemma of Reineke (Exercise 19) without which there would probably be no structure theory for groups of finite Morley rank.
Semisimple Groups
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

We deal here with the structure of semisimple groups and the definition of the generalized Fitting subgroup F* (G) in our context. This is a notion which is used constantly in the analysis of finite simple groups and may be expected to play a parallel role in the context of connected simple groups of finite Morley rank, notably in connection with the solution of Problem C of Chapter B. As the most concrete applications of the general theory given here are to rather small groups, in Chapter 14, we do not yet have solid evidence that the arguments parallel to the finite case go through here; but barring major surprises this notion should turn out to be a workhorse for the further development of the theory. In any case both the Fitting subgroup and the socle in the definable versions used here have proved their utility already, and from that point it is a small step to the generalized Fitting subgroup.
Fields and Rings
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

This chapter may appear as something of a digression, in that it is devoted entirely to the classification of certain rings (e.g. division rings) of finite Morley rank. Actually there is a twofold connection between this material and the analysis of groups of finite Morley rank. In the first place, a ring of finite Morley rank is a group with respect to addition and the multiplication maps are definable endomorphisms. So this allows a direct application of the methods developed in Chapters 4-5. Of course a field (or for that matter a division ring) is essentially a group in two distinct a ways, which gives even more scope for these methods. In the second place there are two powerful methods for finding a field inside a group, one due to Zil’ber in the connected solvable nonnilpotent case and the other to Bachmann in rather more special cases which will arise in Chapters 11 and 13. Knowing that the field is algebraically closed or finite (Theorem 8.1) will sometimes enable us to continue the group theoretic analysis further, notably in Section 9.2.
Geometry and Complex Arithmetic
Authors: Needham Tristan
Published: November 2023

Four and a half centuries have elapsed since complex numbers were first discovered. Here, as the reader is probably already aware, the term complex numberrefers to an entity of the form a+ i b,where aand bare ordinary real numbers and, unlike any ordinary number, ihas the property that i2 = −1. This discovery would ultimately have a profound impact on the whole of mathematics, unifying much that had previously seemed disparate, and explaining much that had previously seemed inexplicable. Despite this happy ending—in reality the story continues to unfold to this day—progress following the initial discovery of complex numbers was painfully slow. Indeed, relative to the advances made in the nineteenth century, little was achieved during the first 250 years of the life of the complex numbers.
Complex Integration: Cauchy’s Theorem
Authors: Needham Tristan
Published: November 2023

In the last few chapters our efforts to extend the idea of differentiation to complex mappings have been amply rewarded. By innocently attempting to generalize the real derivative we were quickly led to the amplitwist concept, and the subject then came to life with a character all its own. While many of the results cast familiar shadows onto the world of the reals, many did not, and striking indeed was the flavour of the arguments used to grasp them. The ability of zto freely roam the plane unleashed in us a degree of visual imagination that had to remain dormant so long as we could only watch the real number xforlornly pacing its one-dimensional prison.
Introduction to self-reference
Authors: Smullyan Raymond M
Published: November 2023

Self-reference plays a crucial role in the famous incompleteness theorem of Gödel[6]. What he did can be roughly described as follows. He showed that for a large class of mathematical systems, one can assign to each sentence a number called the Gödel number of the sentence, and then construct a sentence X asserting that a certain number n has the property of being the Gödel number of a sentence that is not provable in the system, but this number n is the Gödel number of the very sentence X itself! And so X is true if and only if its Gödel number is not the Gödel number of a sentence provable in the system — in other words, X is true if and only if it is not provable in the system. This means that either X is true but not provable in the system, or X is false (not true) but provable in the system. Under the assumption that the system is correct in that only true sentences are provable, the second alternative is out, hence the sentence is true but not provable in the system. This is an extremely rough sketch of what Gödel did.
Some further relations between fixed point properties
Authors: Smullyan Raymond M
Published: November 2023

>We shall now establish some interconnections between single and double fixed point properties. First we will show that if 𝒮 has the recursion property, then 𝒮 also has the weak double fixed point property. [This generalizes results that are known for recursion theory and for combinatory logic.] Suppose 𝒮 has the recursion property; does it necessarily have the double recursion properties DR2, DR1, and DR0? This appears doubtful, but we will show that if 𝒮 is of type 1*, then the answer is yes. Our proof will be divided into two parts as follows.
Extended sequential systems
Authors: Smullyan Raymond M
Published: November 2023

We now turn to a variety of fixed point theorems that simultaneously generalize the fixed point theorems of the last chapter and earlier fixed point theorems for sequential systems. We let 𝒮 be a sequential system (N, Σ, →) such that for every n ≥ 2 there is an exact Σ-function E(x  1  , ..., xn). And now we consider two other items.
Some general incompleteness theorems
Authors: Smullyan Raymond M
Published: November 2023

This chapter is basic to several later chapters of this study, and is devoted to a reconstruction, generalization, and extension of several classical results in mathematical logic concerning incompleteness, provability, and truth. We give abstract versions of the incompleteness proofs of Gödel [6] and Rosser[22] as well as the lesser known incompleteness proof using Tarski’s notion of truth[35]. We conclude the chapter with a fascinating result of Rowitt Parikh[11]. Our results are presented in a purely abstract form which does not employ any apparatus of mathematical logic; the more concrete applications are given in Chapters 5 and 9 of this study.
Self-reference in arithmetic
Authors: Smullyan Raymond M
Published: November 2023

We now turn to the study of self-reference in arithmetic couched in the language of first-order logic. For this chapter and Chapter 9, a prior acquaintance with the logical connectives and quantifiers, though not strictly necessary, is certainly desirable. Part I of this chapter is necessary for the incompleteness proof of Peano Arithmetic given in Chapter 9. The remaining parts of this chapter are not needed for any of the subsequent chapters, but should be of interest to those who wish to see some new and unusual methods of achieving self-reference in arithmetic, as well as a generalization that also generalizes Cantor’s famous proof that a set cannot be put into a 1–1 correspondence with its power set.
Arithmetization of formal systems
Authors: Smullyan Raymond M
Published: November 2023

In Part I of this chapter we prove the fundamental result that if W is formally representable, then the corresponding relation W  0 of dyadic Gödel numbers is Σ1. This has several applications including a neat use of elementary formal systems in incompleteness proofs, which we explain in the next chapter. Next, it leads easily to the crucial fact that every r.e. relation is the existential quantification of a recursive relation — a fact needed for our “double enumeration” of Part II, which is basic to all of Chapters 10 and 11. In Part III we first prove the stronger fact that every r.e. relation is the existential quantification of a Σ0-relation, and thus that the classes of r.e. relations and Σ1-relations are the same. Then we establish some closure properties of recursive relations that are of more specialized interest. Part III is not necessary for any subsequent chapters; the results are of interest in their own right (particularly those of §4).
Elementary formal systems and incompleteness proofs
Authors: Smullyan Raymond M
Published: November 2023

We wish to show here how elementary formal systems can facilitate incompleteness proofs. We avoid use of the usual “Beta-function,” or any of the related devices of Part III of Chapter 8.
Actions of groups
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

We come now to the proper subject of the first part of this book: in this chapter we define exactly what we mean by an action of a group on a set and we introduce some of the appropriate language.
Geometry: an introduction
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

The discussion in this chapter is preliminary to the formal developments which follow. Geometry is in the first instance a description of physical space, and this description becomes algebraic with the introduction of coordinates. Fundamental to our intuition of physical space is the notion of a displacement of a rigid object. Intuitively, a rigid object is one which retains its size and shape when moved about. The property of rigidity can be explained in terms of separation or distance: a physical object is rigid if the separation of any two of its particles is unchanged when the object is displaced. These ideas lead us to the concept of a transformation of the whole of physical space (rather than of an object occupying a piece of physical space) which preserves separation of points. Such a transformation is called an isometry.
The axiomatisation of geometry
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

Classical geometry was presented in the Elements of Euclid in an axiomatic form. that is, as a body of theorems developed by deductive logic from initial postulates. It was, however, what we should now call a physical theory. for it was seeking to describe the spatial relationships of the physical world. The modern notion of an axiomatic theory, which requires only that the axioms shall be logically consistent, did not develop until much later. Thus the possibility of there being more than one ‘valid’ geometry was not envisaged, and was accepted only with difficulty by the learned world in the 19th century when ‘non-Euclidean’ geometries were described.
Projective geometry
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

Although the principles of perspective in pictorial representation were not mastered until c.1500, it is now a matter of common knowledge that two parallel horizontal lines are represented in a picture by two lines which meet on the horizon, or 4Vanishing line’, in the picture (Fig. 13.1). These parallel lines, which do not meet in space, become intersecting lines when projected on the plane of the picture from the point occupied by the eye of the beholder.
Topological considerations
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

The aim of this chapter is to give a brief indication of the way in which topological notions may be brought into play in the study of G-spaces. These ideas are of great importance for further developments in the theory of manifolds, Lie groups and global analysis. Here we can only try to give a preliminary glimpse of the results which flow from the fruitful union of algebra and topology.
Vector geometry
Authors: Roe John
Published: November 2023

In Chapter 1 we explained how Euclid built up the whole of geometry from a small number of fundamental assumptions or axioms. Following Euclid’s method, we stated nine axioms and from them developed the theory of vectors. We defined a vector to be a certain kind of mapping from space 5 to itself: one having the property that for any two points X and Y, the quadrilateral XYv(Y)v(X) is a parallelogram.
Coordinates and equations
Authors: Roe John
Published: November 2023

One of the greatest steps forward in geometry since the time of the Greeks is associated with the name of Rene Descartes (1596-1650). As a philosopher, Descartes was fascinated by geometry because it seemed to offer a model of how certain and reliable knowledge might be obtained.
The Parallel Solution of the Symmetric Eigenvalue Problem
Authors: Barlow Jesse L
Published: November 2023

It is unsound numerical practice to compute A and then find its eigenvalues to obtain the SVD of B since this destroys information. Instead, we use methods which work directly with B.
The exponential sum for the lattice point problem
Authors: Huxley M N
Published: November 2023

This sum arises when we estimate the area under the curve y = NF(x/M) by counting squares, and analyse the error using the row-of-teeth function. The details are given in Chapter 18. We write T /M for N to emphasize the analogy with Chapter 7. The analogy becomes stronger when we make F’(x) in this chapter correspond to F’(x) in Chapter 7, and allow for the effect of binomial coefficients multiplying the rational approximations to derivatives. The reason becomes apparent when we replace F(m) in Chapter 7 by F((m + h)/M)- F(m/M). A corresponding double sum is considered in Chapter 9.
Exponential sums with a difference
Authors: Huxley M N
Published: November 2023

An analogue of the method of Chapter 7 for a general two-dimensional sum does not seem possible, because of the scaling laws. In one dimension, an interval can be divided into N parts, with linear dimensions smaller by a factor 1/N, In two dimensions, the linear scaling factor is only 1/ . This increases the accuracy required from the rational approximations to derivatives of the exponent. But a function of two variables has r +1 partial derivatives of each order r, and the more derivatives to approximate by rationals with a common denominator, the worse the accuracy.
The First and Second Conditions
Authors: Huxley M N
Published: November 2023

First we set up a uniform notation for the Second Spacing Problem in Chapters 7, 8, and 9.
Further results
Authors: Huxley M N
Published: November 2023

The three sections of this chapter are unified only by the idea of iteration. In the course of treating an exponential sum, we come to another exponential sum, or a problem of integer points close to a curve. The first idea is to use the differencing step of Lemma 5.6.2, which produces a double sum with a central difference as in Chapter 9. The end result is slightly weaker than Theorem 17.1.4, but still better than any bound obtained by the van der Corput iteration in one variable. The result would remain weaker than Theorem 17.1.4 even if we had the best possible result in the First Spacing Problem, with Lemma 13.3.1 as sharp as Lemma 13.1.1. These results improve those of Heath-Brown and Huxley (1990), and were partly announced in Huxley (1994b).
Applications to the Riemann zeta function
Authors: Huxley M N
Published: November 2023

The Riemann zeta function, ζ(s) = ∑ 1/n  s, is a generating function for the positive integers. It has a single pole as a function of s, at s = 1 where the series first diverges, because it is the generating function of an infinite set of positive coefficients. It has a functional equation (stated in Section 5.5 of Chapter 5) that relates its values at s and 1 - s; this, in a sense, is equivalent to the Poisson summation formula (Lemma 5.4.2). What makes it extraordinary is that log ζ (s) is a generating function for the prime numbers.
A generalization of Jordan groups
Authors: Adeleke S A
Published: November 2023

For r to be a Jordan set in a group (G, 0) in the usual definition (see p. 73 in the article by Macpherson in this volume) the pointwise stabilizer, G(o,r), must be transitive on r. In this paper we relax this condition and demand that the setwise stabilizer, G(r), be transitive on r. But we also add as assumptions two properties which hold for Jordan sets in the usual definition; namely that the union of any chain of the (new) Jordan sets be a (new) Jordan set, and that the union of any two non-disjoint (new) Jordan sets be a (new) Jordan set. Although infinite simply primitive Jordan groups in the new sense lead to the same G-invariant structures as the old ones of Adeleke and Neumann (in press b), there are simply primitive groups which are Jordan in the new sense but not in the usual sense. Immediate examples are the pathological groups on linear orders described by Glass (1981, Chapter 6, Section 1.10). We make remarks in Section 4 below about the construction of such examples, and it follows easily that there are more examples of the same type built from semilinear orders and C-relations. Apart from including more groups, the result below should be useful in any eventual classification of general infinite simply primitive groups according to their invariant structures. The new Jordan sets and Jordan groups shall be called c-Jordan sets and c-Jordan groups respectively (‘c-Jordan’ as in ‘Camille Jordan’). The proofs of Adeleke and Neumann (in pressb, to appear) need several adjustments to handle the present context.
Recursive saturation
Authors: Evans David M
Published: November 2023

The group Aut 𝓂 of automorphisms of a countable structure 𝓂 (whether considered as a permutation group acting on the domain M of 𝓂, as a topological group, or as an abstract group) only describes the first-order structure of 𝓂 accurately if 𝓂 is a suffiently ‘rich’ or sufficiently ‘saturated’ structure. Considering just one end of the spectrum for the moment, by Scott’s Theorem (p. 20) a countable ℒ”-structure 𝓂 is rigid if and only if every element of 𝓂 is .2”w1w-definable, and there are many rather dissimilar such structures. So one obviously necessary condition for Aut 𝓂 to reflect only first-order information is that 𝓂 be homogeneous (or rather, in the terminology given on p. 20, strongly0-homogeneous), that is that the orbit of any tuple a E M be determined by tp𝓂 (a). But we might also be interested in the first-order properties of tuples of infinite length, subsets of M, equivalence relations on NI, etc., and this would lead us to formulate stronger saturation assumptions concerning 𝓂.
Indiscernibles
Authors: Kaye Richard
Published: November 2023

When investigating the structure of G = Aut(𝓂) as a permutation group acting on M = Dom(𝓂) it is natural to look for subsets I of M for which the induced group
Jordan Algebras
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

In this chapter we introduce Jordan algebras and prove their most fundamental properties. Further developments of an algebraic nature will be presented in later chapters, as needed. We restrict ourselves to finite dimensional algebras over the real or complex field and, except in Section 1, assume the existence of an identity element.
Symmetric Cones And Euclidean Jordan Algebras
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

In This Chapter We Introduce Euclidean Jordan Algebras. As Will Be Seen In Chapter Viii, These are the same as the formally real Jordan algebras more usually considered in the literature. For such algebras we prove a ‘spectral theorem’, which, in the case of the algebra of real symmetric matrices, specializes to the usual spectral theorem. Then we prove a fundamental result which establishes a one-to-one correspondence between symmetric cones and Euclidean Jordan algebras. A consequence of this is the unique decomposition of every symmetric cone into a direct product of irreducible ones. Finally, for symmetric cones, we give an independent description in Jordan algebra terms of all the geometric and analytic objects introduced in Chapter I.
Wave forces on offshore structures
Authors: Rahman Matiur
Published: November 2023

The determination of the forces induced by water waves on an offshore structure is one of the central interests in the design of the structure. There is a large variety of offshore structures which can be described as being composed of small tubular members, large gravity platforms, tension-legged platforms, with large vertical, horizontal, diagonal, circular or square cylinders, semi-submersibles and arctic structures. These structures may be fixed and compliant and also may be floating in waves. Whatever the case, the complexity of the interaction of waves with the structures makes the calculation of wave forces more difficult. Furthermore, owing to the inherent nonlinear nature of ocean waves, no suitable nonlinear mathematical theory is available at present to predict the wave forces on an offshore structure, the following three methods are available to calculate wave forces: (a) the Morison equation, (b) Froude-Krylov theory and (c) diffraction theory. The distinction between these methods is described below.
Discretizing And Periodizing The Half-Plane
Authors: Holschneider M
Published: November 2023

Because of the reproducing kernel identity the wavelet coefficients have an internal correlation and are therefore not independent. So it is not surprising that knowledge of the wavelet transform on a subset of the half-plane is sufficient to reconstruct the analysed function. In the following sections we consider various partial reconstruction formulas associated with different subsets of the half-plane. In particular, we consider reconstruction over strips, over zooms and cones, and over various grids. In addition, we will construct a wavelet analysis over the circle. For the sake of simplicity we will formulate everywhere in one dimension. The reader should have no difficulty in generalizing the results to wavelet transforms of functions over ℝn.
Group Theory As Unifying Language
Authors: Holschneider M
Published: November 2023

In this chapter we will give a very short introduction to the theory of representations of groups and their application to the understanding of the wavelet transform, at least in a Hilbert-space context. The discrete wavelet transform can be understood in an algebraic setting.
The Hecke Ring Of GlnOver A Local Field
Authors: Macdonald I G
Published: November 2023

Let F be a locally compact topological field. We shall assume that the topology of F is not the discrete topology, since any field whatsoever is locally compact when given the discrete topology. A non-discrete locally compact field is called a local field.
Zonal Polynomials
Authors: Macdonald I G
Published: November 2023

For a subgroup K of G, the following conditions are equivalent: (a) the induced representation 1G  K  is multiplicity-free: (b) the algebra C(G, K) is commutative If these equivalent conditions are satisfied, the pair (G, K) is called a Gelfand pair.
More specific results on critical points
Authors: Rahman Q I , Schmeisser G
Published: November 2023

Let fi be a polynomial of degree nj, having all its zeros in the interval [aj,bj], where j 1, 2 and a1 < b1 < a2 < b2. If f1 (b1) h(a2) = 0, then, by Rolle’s theorem, the product f : z ↦ f1 (z) f2 (z) has exactly n1 ™1 critical points in [a1,b1], one critical point in (b1,a2), and n2 -1 critical points in [a2,b2]. When the zeros of f1 and f2 do not lie on two disjoint segments of a straight line, we may apply the Gauss Lucas theorem to conclude that, if fi is the set of the zeros of fj, where j = 1, 2, then the critical points of f lie in the convex hull of ε1 Uε2. The following theorem of Walsh shows that, in this general case, we can say more in terms of circular domains 𝒦1, 𝒦2 containing the sets ε1, ε2, respectively.
Inclusion of all zeros
Authors: Rahman Q I , Schmeisser G
Published: November 2023

In the following sections, most of the bounds for the zeros of a polynomial f are given in terms of a zero p of an associated auxiliary polynomial. As such, these bounds are not explicit, but as we shall see, they are often the best possible in some sense. We shall also deduce various simple explicit bounds for this zero p. In addition, p can be easily approximated by numerical methods.
Sampling Theory and the Arithmetic Fourier Transform
Authors: Walker W J
Published: November 2023

The Arithmetic Fourier Transform (AFT) is an algorithm for the computation of Fourier coefficients which was given this name by Tufts and Sadasiv (1988). The algorithm has a long history differentiated by three distinct stages, the first of which began with the astronomers of the nineteenth century. At that time the calculation of the periodicities of the motions of planets or of tidal effects involved extremely large amounts of empirical data. The German mathematician H. Bruns (1903) published a method for the computation of the Fourier series of a periodic function by using Mobius inversion. This algorithm utilized the magnitude of the data available and was constructed using alternating sums of sample values.
Abstract Harmonic Analysis and the Sampling Theorem*
Authors: Dodson M M , Beaty M G
Published: November 2023

The step from Fourier analysis to abstract harmonic analysis is a natural one which unites a number of apparently disparate results into one general framework with a concise and elegant notation. The classical sampling theorem is a good example of this, since a number of different versions can be obtained from the one abstract result due to I. Kluvanek (1965) simply by choosing the appropriate underlying locally compact abelian group. J.R. Higgins (1985) gives several examples where the sampling theorem is obtained for the real line ℝ, the circle 𝕋, the torus 𝕋, the integers ℤ, the dyadic group and Euclidean space ℝ  n,. Haar measure, a translation-invariant measure associated with the locally compact abelian group, plays a prominent role in the abstract theory; reciprocity relations and measure relations between the Haar measures of certain groups and sets emerge as natural and useful. The coset decomposition formula (10.2.3) below plays a particularly important part as it allows the analysis to be carried out in the space of square-integrable functions on a compact abelian group.
Amalgams
Authors: Pasini Antonio
Published: November 2023

We will define amalgams in Section 11.2 of this chapter, aiming at a construction of universal objects in categories of parabolic systems (we post pone the more general problem of constructing universal objects in categories of geometries to the next chapter). Then we apply amalgams to the investigation of flag-transitive thin geometries (Section 11.3).
Cn and F4
Authors: Pasini Antonio
Published: November 2023

This chapter is a continuation of the last section of Chapter 13.

In this section r is a geometry of type C3. We do not assume that r is finite or thick-lined, for the moment.
Integrability, Self-Duality, and Twistor Theory
Authors: Mason L J , Woodhouse N M J
Published: November 2023

It has been known for some time that many of the familiar integrable systems of equations are symmetry reductions of self-duality equations on a metric or on a Yang-Mills connection (for example, the Korteweg-de Vries and nonlinear Schrödinger equations are reductions of the self-dual Yang-Mills equation). This book explores in detail the connections between self-duality and integrability, and also the application of twistor techniques to integrable systems. It has two central themes: first, that the symmetries of self-duality equations provide a natural classification scheme for integrable systems; and second that twistor theory provides a uniform geometric framework for the study of B¨acklund tranformations, the inverse scattering method, and other such general constructions of integrability theory, and that it elucidates the connections between them.
Introduction
Authors: Mason L J , Woodhouse N M J
Published: November 2023

It is easier to give examples of ‘integrability’ than to say precisely what it means. The mathematical literature contains an impressive body of well-developed theory, but no single effective characterization that covers all the known cases. The difficulty is that the theoretical development has been driven very largely by the study of particular examples, and that the mathematical tools that are used to deal with one class of examples do not always carry over to another. As the catalogue of theory and example grows, older definitions that capture the exact meaning of integrability in finite-dimensional systems are seen to lack precision and universality.
ASD metrics
Authors: Mason L J , Woodhouse N M J
Published: November 2023

In this chapter, we shall look at various forms of the ASD condition on the curvature of a metric in four dimensions. The conditions are naturally expressed in terms of ‘Lax pairs’ of vector fields, either on space-time itself or on some bundle over it. By interpreting a vector field as a generator of a diffeomorphism group, we can represent some of the conditions as special cases of the ASDYM equations, but with infinite-dimensional gauge groups. In §13.1-4 we derive various forms of the ASD condition and its subcases on the curvature of a metric. In §13.5 we describe the extension of the twistor correspondence by which ASD conformal structures are encoded in a curved twistor space P, and the way in which various supplementary conditions on the metric (vacuum, Einstein, Kāhler) give rise to additional structures on P. In §13.6 we discuss the reductions by one symmetry to the equations of an Einstein-Weyl space and to the ‘SU(∞) Toda field equations’. In §13.7 we shall explain how many standard integrable equations arise as reductions of the ASD conditions on a metric, and how the curved twistor construction relates to the Penrose-Ward transform.
New groups from old
Authors: Leedharn-Green C R , McKay S
Published: November 2023

This chapter contains further basic results and definitions in group theory that will be needed, concentrating on methods of constructing groups. Sections 2.1 and 2.2 are concerned with extensions, pull-backs and central products: and Sections 2.:3 and 2.4 with wreath products. Again standard results are often quoted without proof, and can be found in (Huppert 1967).