bluuebunny/crossref_metadata_2025_split huggingface dataset:
Dynamical Weyl Group
Authors: Etingof Pavel , Latour Frédéric 
Published: November 2023

Abstract
In this chapter we will review the theory of the dynamical Weyl group. This theory has many applications, in particular to study of traces of intertwining operators. The theory of the dynamical Weyl group was developed in Tarasov and Varchenko (2000), Etingof and Varchenko (2002).
The stars on the astrolabe
Authors: Kunitzsch Paul 
Published: November 2023

Abstract
The Greek word ‘astrolabe’ means ‘star catcher’. This name exactly describes an essential element of the practical use of the instrument. By means of a sighting device – called the ‘alidade’ (Arabic al-cicada ) – which is placed on the back of the instrument, one can measure the altitude of a heavenly body above the horizon. Then, the position of a star, the Sun, or the Moon can be set by turning the ‘rete’ (‘net’ or ‘spider’; Arabic al-cankabut , Latin alhancabuth , etc.) located on the front of the instrument. The rete is a perforated plate which represents a two-dimensional map of the heavens. The main structural elements of the rete include the ring of the ecliptic and one or more supporting bars which are used to connect the elliptical ring to the circumference of the plate. A number of pointers are attached to these structural elements.
Modal Logic And Closure Algebras
Authors: Dunn J Michael , Hardegree Gary M 
Published: November 2023

Abstract
The reader who has no previous familiarity with modal logic may wish to consult Hughes and Cresswell (1968, 1996). The classic, but now dated reference, is Lewis and Langford (1932). The modal logics which we shall consider are only a few among many. The reader wanting to know more about the many should consult Chellas (1980), Segerberg (1971) or Gabbay (1976). Incidentally, this section is very much influenced by Lemmon (1966) and Lemmon with Scott (1977).
Representations And Duality
Authors: Dunn J Michael , Hardegree Gary M 
Published: November 2023

Abstract
The study of representations of lattices and related structures has been considerably enriched since Stone ‘s original work, with contributions from various authors. We restrict ourselves here to a brief report for the sake of the interested reader. We begin by bringing attention to an important aspect of the Stone representation, called “duality, “ that we did not previously discuss. Let us first try to give philosophical meaning to duality issues. Duality gets its ultimate expression in the terms of category theory, but we shall do our best to give the reader some intuitive grasp of duality without resorting to category theory, or at least its diagrams.
Algebraization
Authors: Kawakubo Katsuo 
Published: November 2023

Abstract
We have studied transformation groups in a geometric way. In this chapter we change entirely the method employed so far and consider, so to speak, algebraization of geometric structures. Specifically, we shall study the algebraic aspects of transformation groups. The essence of transformation groups will be abstracted, recognized, and understood via algebraization. As a means of algebraization, we introduce the concept of G-complexes for G, a finite group, and study first the relationships among the Euler characteristics of Gcomplexes, orbit spaces, and various kinds of fixed point sets. The Mobius inversion formula plays an important role in this study. When G is a compact Lie group, similar formulae for Euler characteristics of G∼manifolds are also obtained by making use of the decomposition theorem in Section 4.6 (see Section 5.6).
Trees
Authors: Read Ronald C , Wilson Robin J 
Published: November 2023

Abstract
A tree is a connected graph that has no cycles. If a tree has n vertices, then it has n − 1 edges, and any tree with more than one vertex has at least two end-vertices (vertices of degree 1). In any tree, there is a unique path between any two given vertices, and the removal of any edge disconnects the tree.
A rooted tree is a tree in which a particular vertex, the root, has been singled out. A tree is homeomorphically irreducible if it has no vertices of degree 2. An identity tree is a tree whose only automorphism is the identity map—that is, a tree with no non-trivial symmetries. For example, tree (a) below is homeomorphically irreducible, and tree (b) is an identity tree. A binary tree is a rooted tree in which at each vertex there can be a left upward branch or a right upward branch, or both or neither (see tree (c)).
Regular Graphs
Authors: Read Ronald C , Wilson Robin J 
Published: November 2023

Abstract
A graph is regular if all its vertices have the same degree; in particular, graphs that are regular of degree 3, 4, 5 and 6 are called, respectively, cubic, quartic, quintic and sextic. An example of each type is shown below.
A regular graph is polyhedral if its vertices and edges correspond to the vertices and edges of a 3-dimensional polyhedron-that is, it is planar and 3-connected. A bicubic graph is a graph that is both bipartite and cubic. The following polyhedral graphs are both cubic, but only the first is bicubic.
Types Of Graph
Authors: Read Ronald C , Wilson Robin J 
Published: November 2023

Abstract
Recall that a graph G is bipartite if its vertex set V can be partitioned into two sets X and Y so that each edge joins a vertex in X and a vertex in Y, and that G is Hamiltonian if it has a cycle that includes each vertex exactly once. G is an even graph if each vertex has even degree; thus, a connected even graph is an Eulerian graph. G is self-complementary if G is isomorphic to 1”5 complement; the number of vertices in a self-complementary graph must be of the form 4r or 4r + 1, where r is an integer. G is triangle-free if it has no cycles of length 3, and unicyclic if G contains only one cycle.
Signed Graphs
Authors: Read Ronald C , Wilson Robin J 
Published: November 2023

Abstract
A cycle in a signed graph is a positive cycle if the number of negative edges is even and is a negative cycle if the number of negative edges is odd. A signed graph is balanced if and only if each cycle is a positive cycle, and unbalanced otherwise; equivalently, a signed graph is balanced if we can colour each vertex red or green in such a way that positive edges have ends of the same colour and negative edges have ends of different colours. Signed graphs are used in the social sciences, with positive edges corresponding to positive relationships and negative edges corresponding to negative ones. Balanced signed graphs correspond to social situations in which there is no ‘tension’.
Ramsey Numbers
Authors: Read Ronald C , Wilson Robin J 
Published: November 2023

Abstract
The (diagonal) Ramsey number r(G) of a single graph G is the Ramsey number r(G, G). On pages 374—379 we depict all isolate-free graphs with up to 7 edges, with their Ramsey numbers. The R-number appearing in each top-right comer is the serial number of the graph in Burr’s catalogue; a graph that appears elsewhere in this Atlas is cross-referenced in the top- left corner. On page 380 we depict the graphs with more than 7 edges for which the Ramsey number is known.
A selection of elementary problems treated by eminent twentieth-century mathematicians
Authors: Gofman Judita 
Published: November 2023

Abstract
What kinds of research work are professional mathematicians concerned with nowadays? Answers to this question are beyond the scope of our book. Nevertheless, we shall be able to discuss here a variety of elementary, beautiful and intriguing problems which have caught the attention of eminent contemporary scholars. In addition to being intrinsically interesting, the problems chosen for this chapter are connected with important branches of modern mathematics.
Project 4
Authors: Grandine Thomas A 
Published: November 2023

Abstract
Because of his newly found confidence in your ability, Biro Kratt has just assigned you to a more challenging project. It appears that the ACME Super Power Usefully Transmitting Nice Information computer satellite has begun to malfunction. Your boss swears that it’s the last piece of high-tech equipment he will ever buy at a rummage sale. However, the malfunction may still be dealt with, and you have been assigned the task of overcoming the difficulty. The nature of the malfunction is that 80 per cent of the signal is getting lost in static. Your job is to try to reconstruct the signal based on the 20 per cent of the signal which is getting through. You will attempt to do this using interpolation.
Project 7
Authors: Grandine Thomas A 
Published: November 2023

Abstract
The Corporate Headquarters for Extraneous Accounting Procedures, an administrative group within ACME, needs to make an important business decision. CHEAP wants to borrow $100 million in order to finance an important research project whose ultimate success will prove extremely beneficial for ACME. CHEAP has located two potential sources for this money. The first, Mutual United Stock and Corporate Liabilities Exchange, is willing to loan the $100 million at an annual interest rate which has not yet been negotiated. CHEAP would be expected to repay the loan over 10 years by making ten equal payments to MUSCLE at the end of each year. The other source of money is the Security Home Annuity Refund Kartel.
Introduction: logic and language
Authors: Potter Michael D 
Published: November 2023

Abstract
In principle the exposition of a first-order mathematical theory need consist of no more than a list of assumptions, called axioms, followed by a list of assertions, called ,*theorems each of which is written in the language of the theory and follows from the axioms and the previous theorems by means of the rules of classical logic. Checking such a list of theorems for deductive correctness would be a very boring but entirely mechanical process and it would be easy to program a computer to do it. However, such a list would be quite useless as a means of communicating mathematics from author to reader and we shall not provide one here. What follows should be regarded instead as an informal specification of such a list: theorems will be stated in a mixture of the formal first-order language and informal English; we shall from the outset use definitions to introduce abbreviations, without which the statements of even the simplest theorems would be far too long to take in; and the ‘proofs’ will in fact be no more than attempts to indicate how a fully formalized proof could be constructed.
The axiom of choice
Authors: Potter Michael D 
Published: November 2023

Abstract
We return now to the consideration of the conception of set theory which our axiom system is intended to codify. We have already (in the note on ‘Indefinable collections’ in §1.2) remarked on the question whether we should allow the possibility of collections which are not, even in principle, definable. We left this question unresolved then, but in §5.5 we were forced, in proving that every set is finite or infinite, to assume the existence of a sequence which we could not explicitly define. The axiom which we used there-that of countable choice-is the prototype for the ones we shall be considering here. In particular, the assertion which we obtain by removing from it the assumption of countability-the so-called ‘axiom of choice’-is central in what follows.
The prime ideal property
Authors: Potter Michael D 
Published: November 2023

Abstract
In this chapter we shall show that (assuming the axiom of choice) every small distributive lattice L can be represented as (i.e. is isomorphic to) a sublattice of an appropriately chosen power set 𝒟 (spec(L)). This generalizes the results on the representation of spatial lattices which we obtained in §8.7. In the course of this endeavour we shall come across several lattice-theoretic assertions provably equivalent to the axiom of choice, and several (including the representation result itself) equivalent to a weaker principle called the ‘prime ideal property’.
Product operators and others
Authors: McLarty Colin 
Published: November 2023

Abstract
This chapter deals with a formal issue. With products, for example, once we know that objects A and B have at least one product we pick one and call it A × B. Similarly, if B has at least one exponential by A we pick one and call it B’ As long as we deal with only a finite number of objects at a time this can be formalized in first order logic, with no axiom of choice, using no actual terms A × B or B. But if, for a given A, we want to once and for all pick one exponential BA for each B, that involves an axiom of choice. And even if only a finite number of choices are made there is a kind of awkwardness in them. There fore we adopt a device for avoiding them.
Mathematical foundations
Authors: McLarty Colin 
Published: November 2023

Abstract
The axioms for, say, a Cartesian closed category are a sufficient foundation by themselves for the theory of a single such category. No stronger background theory or metatheory is needed for the theorems of Chapter 6. Similarly, the axioms in Chapter 22 suffice as a formal foundation for a category of sets, and those in Chapter 23 for a category of smooth spaces. But the past few chapters have worked with many categories at once, and this requires some further foundation. In what follows, two styles of mathematical foundations for general category theory are described. The first, found in most of the literature, interprets category theory in some axiomatic set theory. The second, still being developed, axiomatizes a category of categories in categorical terms.
Euclidean rings
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
The division algorithm for integers is enormously useful, and so it makes sense to investigate whether something similar and equally useful works, say, for Gaussian integers. When it works (as indeed it does for Gaussian integers), it can be used as the foundation for a successful extension, to the rings in question, of arithmetic ideas such as prime numbers and highest common factors. It is then possible to use these new arithmetics to solve problems in classical number theory and some other branches of mathematics. We shall give some of these applications in later chapters.
The four-squares theorem
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
We are now in a position to use the ring of Gaussian integers to prove one of the most famous theorems in number theory: every positive integer is the sum of four squares. The material of this chapter is a bit tricky and is not used later, so that the reader may omit it or read it superficially without worrying about the details. Nevertheless, we recommend that you look at Comments 4.7 and the Exercises. Let us set the scene by considering some typical problems of classical number theory. Because for the time being we are in number-theory mode, when we say ‘square’ we mean ‘square of non-negative integer’ and similarly for cubes, etc. There has been a great deal of interest in questions about whether every positive integer can be represented as the sum of such-and-such a number of special integers; or which integers can be represented in this way; or can every large enough integer be represented in this way. Here are some examples.
Unique factorization domains
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
At first it is perhaps rather difficult to appreciate why so much importance is placed on the result that every positive integer (except 1) is a unique product of prime numbers (see 3.14). It is known as the fundamental theorem of arithmetic, which suggests that it must be very significant, but in practice we take it for granted that, for instance, the only way of writing 10 as a product of prime numbers is as 2·5 (or 5 · 2, but that is really the same thing). We might hope that this sort of unique factorization property, when suitably defined, also holds in such systems as the Gaussian integers. If it does, we will have to prove it; we have no right to take it for granted. In fact we shall show that unique factorization does hold for Gaussian integers, but we shall also give examples (one of them apparently rather similar to the Gaussian integers) where unique factorization fails. It is these examples of failure which may bring home to us for the first time the need to be careful. Unique factorization should not be taken for granted, and when it does hold we should appreciate its value in underpinning some of the arithmetic arguments which we may want to use.
Algebraic numbers
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
Many important numbers, such as ../2 and i, are solutions of polynomial equations. Such numbers are said to be algebraic, and we shall study their basic theory in this chapter. Algebraic numbers also occur frequently in the following context: We start with a polynomial f(X) over Q and we wish to study the solutions of the corresponding equation f(X) = 0. These solutions are algebraic numbers. Definition 11.l. Let K be a field with a subfield F. Then t EK is said to be algebraic over F if f(t) = 0 for some non-zero polynomial f(X) with coefficients in F.
Ruler-and-compass constructions
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
In the Greek tradition of geometry there has been much interest in determining which geometrical constructions can be performed using only two very simple devices. This question is asked in an idealized context where constructions can be performed with complete accuracy. The first permitted device is a ruler or straight edge; this makes it possible to draw the line joining distinct points A and B in the plane and to extend the line as far as necessary in either direction. The ruler has no markings on it and it cannot be used for measuring lengths. The second device is a pair of compasses; this makes it possible to draw the circle with centre at given point A passing through a second point B. It was known to the ancient Greek geometers that it is possible, using only these devices in combination, to perform such constructions as bisecting a given angle or dropping a perpendicular from a point to a line (details below).
Toposes - Basic Theory
Authors: Johnstone Peter T 
Published: November 2023

Abstract
We are now ready to present the definition of an (elementary) topos, which is the central notion of this book: it is a category having all the good ‘set-like’ properties discussed in the previous chapter, except (possibly) for Booleanness. It is a remarkable fact that all these properties are implied by the conjunction of just two of them:
Allegories
Authors: Johnstone Peter T 
Published: November 2023

Abstract
The object of this Chapter is to present an alternative approach to the notion of topos, different from (but not entirely independent of) the one which we followed in Chapters A1 and A2, via categories of relations. Whlle this appt'cach will not play a very important role in our subsequent exposition of topos theory, it is of some interest in its own right, and it certainly provides an appealing route to the construction of some of the leading examples of toposes, which can make their structure easier to understand.The advantages of this approach have been forcefully argued over some years by P. Feeyd, and a fuffer account of it wlll lee found in [381]. There are a number of other approaches to the study categories of relations - notably that developed by A. Carboni and ft. Wallers [322], whidi makes heavier use of 2-categorical ideas - but we shall not describe them here.
Many-particle systems
Authors: Lunn Mary 
Published: November 2023

Abstract
There are many binary systems of stars studied by astrophysicists. The binary system is particularly stable but true tertiary star systems are relatively rare. In fact tertiary systems often comprise a binary system together with a single star. There is no complete analytical solution to the equations satisfied by a tertiary system of star. For systems of more than two stars the equations have to be solved by numerical methods and stable solutions appear to be rare. The planetary system is of course stable but then the Sun is considerably more massive than any of the planets. We shall study the equations associated with many-particle systems very briefly.
An Introduction to Spectral Methods for Partial Differential Equations
Authors: Quarteroni A 
Published: November 2023

Abstract
This lecture is an elementary introduction to spectral methods for the numerical approximation of boundary value problems. The basic properties of Fourier and Chebyshev orthogonal expansions are revisited. The discrete Fourier and Chebyshev transforms are introduced, and it is shown how to use them in the framework of spectral collocation approximations. Several examples of applications are shown, including elliptic boundary value problems, advection-diffusion equations and Navier-Stokes equations. Both periodic and non-periodic boundary conditions are considered.
Univariat Splin S
Authors: Dierckx Paul 
Published: November 2023

Abstract
Spline functions (splines) are currently used in diverse domains of numerical analysis (interpolation, data smoothing: computer aided geometric design. the mmwrical solution of differential and integral equations, etc.). A landmark iu their increasing popularity has certainly been the publication of efficient algorithms for calculating with B-sp1ines (de Boor 1972; Cox 1972). The latter still form the basis of several high quality spline software packages.
In this first chapter we will collect the basic properties of splines and B-splines together for further reference. We will intuitively explain why the latter are so spf ‘cial and important. We will also give an impression of the computational complexity of the B-spline algorithms mentioned above. Exact proofs will be omitted. They can be found in the basic works of de Boor (1978) and Schumaker (1981).
A more recent spline topic is knot insertion. It plays a central part in interact in• CAD spline systems. It will also appear to be extremely useful in a numht&gt;r of applications that we will consider. \Ve will therefore briefly commt&gt;nt on knot insertion algorithms and indicate some connections with the other B-spline algorithms.
Curve Fitting: An Introduction
Authors: Dierckx Paul 
Published: November 2023

Abstract
In this introductory chapter we will state the curve fitting problem and briefly review a number of published spline solution methods. Reference will be made to particular software implementations of which we are aware. They are to be found in the appropriate chapters of the well known general software libraries (NAG (1988), IMSL (1987), PORT (Fox 1984), HARWELL (1987)) or in specific spline packages (fitpack (Cline 1974) and FITPACK ; Dicrekx 1987a. 1989). not to be confused, DASL (Anthony and Cox 1987b), SPLINE TOOLBOX for use in MATLAB (de Boor 1990)). A lot of these routines are also distributed freely through netlib (Dongarra and Grosse 1987); readers with an electronic mail facility are advised to consult the netlib libraries dierckx, fitpack, gcv, pppack and toms in particular (email address: netlib@oml.gov). Finally, we also refer to the paper by Grosse (1990) which contains an extended catalogue of algorithms for numerical approximation with references to the literature and pointers to available code