bluuebunny/crossref_metadata_2025_split huggingface dataset:
Homomorphisms, ideals, and factor rings
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
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.
Finite fields
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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).
Turing machines
Authors: Howie John M 
Published: November 2023

Abstract
"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.
Basic Group Theory
Authors: Borovik Alexandre , Nesin Ali 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
&gt;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

Abstract
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

Abstract
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

Abstract
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

Abstract
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).
Geometry: an introduction
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
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

Abstract
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.