bluuebunny/crossref_metadata_2025_split huggingface dataset:
Projective geometry
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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.
The exponential sum for the lattice point problem
Authors: Huxley M N 
Published: November 2023

Abstract
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

Abstract
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.
Further results
Authors: Huxley M N 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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.
Symmetric Cones And Euclidean Jordan Algebras
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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.
More specific results on critical points
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
Let fi be a polynomial of degree nj, having all its zeros in the interval [aj,bj], where j 1, 2 and a1 &lt; b1 &lt; a2 &lt; 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

Abstract
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

Abstract
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

Abstract
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.
Integrability, Self-Duality, and Twistor Theory
Authors: Mason L J , Woodhouse N M J 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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).
p-groups of maximal class
Authors: Leedharn-Green C R , McKay S 
Published: November 2023

Abstract
A group of order pn and nilpotency class n − 1 that is, of coclass 1, is said to be of maximal class if n &gt; 4. The basic material about these groups is to be found in (Blackburn 1958)1 and is also covered in the chapter on finite p-groups in (Huppert 1967). Around 1970 various authors, perhaps prompted by read ing this account of the theory, took up the study these groups and substantial advances were made in understanding their structure (Shepherd 1971; Miech 1970, 1974a,b; Leedham-Green and McKay 1976b, 1978a,b, 1984).
Pro-p-groups
Authors: Leedharn-Green C R , McKay S 
Published: November 2023

Abstract
This chapter begins the investigation of the connection between pro-p-groups of coclass r and finite p-groups of coclass r. The first three sections of the chapter contain the basic results and definitions in the theory of pro-p-groups that will be needed, beginning with more general topological groups. Again standard results are often quoted without proof, and can be found in (Pontrjagin 1957) or in (Dixon et al. 1999). See also (Wilson 1998).
Connections
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
In this chapter we begin our study of Yang-Mills theory—the theory of connections-which makes up the core of this book. Classical differential geometry considers, by and large, connections on the tangent bundle of a manifold, the most important being the Levi—Civita connection defined by a Riemannian metric. The key feature of Yang—Mills theory is that it deals with connections on auxiliary bundles, not directly tied to the geometry of the base manifold. After reviewing standard definitions and notation in Section 2.1, we go on to give proofs of two fundamental theorems: the integrability theorem for bolomorphic structures and Uhlenbeck’s theorem on the existence of Coulomb gauges. These are essentially local results. We regard them as extensions of the elementary fact that connections with curvature zero are locally trivial.
Yang-Mills Moduli Spaces
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
In this chapter we introduce moduli spaces of ASD Yang-Mills connections and develop some of their basic properties. Let E be a bundle over a compact, oriented, Riemannian four-manifold X. At the level of sets, the moduli space ME is defined to be the set of gauge equivalence classes of ASD connections on E: that is, we identify connections which are in the same orbit under the action of the bundle automorphism group 𝒢 The moduli space has a natural topology, induced by the topology in the space 𝒜 of connections.
Stable Holomorphic Bundles Over KÄhler Surf Aces
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
In this chapter we consider the description of ASD moduli spaces over complex KÄhler surfaces. This discussion takes as its starting point the relation between the ASD equation and the integrability condition for § operators which we have seen in Chapter 2. We show that the ASD connections can be identified, using a general existence theorem, with the holomorphic bundles satisfying the algebro-geometric condition of ‘stability’. Examples of concrete applications of the theory have been given in Chapter 4; and in Chapters 9 and 10 we will see how these ideas can be used to draw conclusions about the differential topology of complex surfaces.
Non-Existence Results
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
In this chapter we return at last to address some of the problems in fourmanifold topology which were raised in Chapter 1. We will prove Theorems (1.3.1) and (1.3.2) on the non-existence of manifolds with certain intersection forms. Throughout the chapter, the strategy of proof of these non-existence results will be the same. Supposing that a manifold X of the type in question exists. one chooses judiciously a bundle E over X and studies the moduli space of ASD connections M=ME. From the known topological features of the moduli space and its embedding in the space ℬE one derives a contradiction.
Generalized Riemann And Variational Integration In Division Systems And Division Spaces
Authors: Henstock Ralph 
Published: November 2023

Abstract
My thanks are due to Thomson (1971a, b, c, 1972a, b, 1975), who examined the theory and showed that it would work with fewer hypotheses, coining the name, division system, which in this book is called a free division system. Here I have called a division system one that considers divisions of a fixed elementary set. See Example 1.1.3 for some details of Thomson’s system.
Integration In Infinite Dimensional Spaces
Authors: Henstock Ralph 
Published: November 2023

Abstract
In countable Cartesian product spaces (sequence spaces) a result of Jessen (1934) holds, so that the integral is the limit of the corresponding integral over the first n terms of the sequences. The origins of integration in uncountable Cartesian product spaces (function spaces) are in Einstein (1906) and Smoluchowski (1906), examples being given by Chandrasekhar (1943). Two integrals vital for theoretical physicists and theoretical chemists are in Wiener (1930) and Feynman (1948). It may be that the integral of this chapter gives mathematical expression to the latter; other expressions used are far too complicated.
Introduction
Authors: Dwork Bernard 
Published: November 2023

Abstract
We examine the Boyarsky principle for generalized hypergeometric functions. This involves understanding the integral representations of such functions to the point of being able to show that the Frobenius matrix varies analytically with the multiplicative parameters (e.g. a, b, c in the case of the Gauss 2F1(a, b, c, λ). Following a suggestion of G. Laumon (letter of 25 April 1983) we use the Laplace transform to develop a theory broad enough to encompass the four hypergeometric functions of Appell (Appell and Kampe de Feriet 1925), their Lauricella generalizations and kFk-i (Erdelyi et al. 1953). Even in the case of 2F1 the present work represents a significant improvement over our previous study (Dwork 1983) of the Boyarksy principle. From the point of view of cohomology associated with hypergeometric functions our work represents an improvement over that of Aomoto (1975, 1977) who seems to have restricted his attention to cyclic coverings of ramified at a set of non-singular hypersurfaces with normal crossings (i.e. the intersection of each subset is a non-singular complete intersection). The condition is satisfied by 2F1, F; of Appel and the Lauricella FD but not by the remaining hypergeometric functions listed above.
Deformation Theory
Authors: Dwork Bernard 
Published: November 2023

Abstract
We use the notations of Chapters 2 and 3 except that here λ= (λ1, …, λm) consists of m algebraically independent elements over ℚ(π, a). We put B = ℤ[λ], K = ℚ(a, π, λ) and completion will play no role. We consider satisfying (2.6).
Singularities (Generalities)
Authors: Dwork Bernard 
Published: November 2023

Abstract
We now discuss the singularities of eqn (5.17) . To clarify the notation we emphasize the tangent space aspect of the deformation mapping. In this direction let a denote a generic point on H over ℚ, let λ = (λ1, …, λm) be a generic point in affine m-space and Γ a generic point on the affine line.