bluuebunny/crossref_metadata_2025_split huggingface dataset:
The structure of finite p-groups
Authors: Leedharn-Green The structure of finite p-groups [CT] C R , McKay S 
Published: November 2023

Abstract
The general aim of this chapter is to prove that if P is a finite p-group of coclass r then P has a normal subgroup N of order bounded by a function of P and r such that P/N is constructible as defined in Definition 8.4.9. This generalizes the theory of P-groups of maximal class described in Section 8.2; in particular, it generalizes the observation that if Pis a 2-group of maximal class and order at leru;t 16 then P ha.-, a normal subgroup N of order 2 such that P/N is dihedral. We give generic bounds in terms of P and r to the order of N and the derived length and other invariants of P.
Beyond coclass
Authors: Leedharn-Green C R , McKay S 
Published: November 2023

Abstract
The primary step in classifying the finite p-groups of coclass at most p lies in the classification of infinite pro-p-groups of coclass at most r. The question then arises as to what other classes of pro-p-groups can be used for finding other classifications of finite p-groups. At this point the pro-p-groups become at least as interesting as the finite p-groups. The conditions to be imposed on the pro p-groups are generally finitistic, and hence apply vacuously to finite p-groups.
Four-Manifolds
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
This chapter falls into three parts. In the first we review some standard facts about the geometry and topology of four-manifolds. In the second we discuss a number of results which date back to the 1960’s and before; in particular we give an account of a theorem of Wall which accurately portrays the limited success, in four dimensions, of the techniques which were being used to such good effect at that time in the study of high-dimensional manifolds. This discussion sets the scene for the new developments which we will describe in the rest of this book. In the third section we summarize some of the main results on the differential topology of four-manifolds which have sprung from these developments. The proofs of these results are given in Chapters 8, 9, and 10. The intervening chapters work, with many digressions, through the background material required for these proofs.
Topology And Connections
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
In Section 5.3 we discuss a different route by which the topologies of X and ℛ* are related: this is through the notion of a ‘concentrated’ or ‘particle-like’ connection, whose curvature is concentrated near a finite collection of points. We have seen in Section 4.3 that such connections can be expected to arise near the boundary of the moduli space of ASD connections. Here we shall examine the topological content of this phenomenon, and its relationship to the Poincare duality pairing on X.
The Differential Topology Of Algebraic Surfaces
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
Let us now review our position with regard to the overall aims of this book. In Chapter 1 we set up the twin problems in four-manifold theory: deciding the existence and uniqueness of manifolds with given intersection forms. In Chapter 8 we have shown how arguments with Yang-Mills moduli spaces give strong non-existence results, and in Chapter 9 we have developed invariants of four–manifolds aimed at the complementary question of uniqueness. However we have not yet given any full-blooded example of the application of these invariants, i.e. shown that they can distinguish differentiable four-manifolds with the same intersection form, and this is the purpose of the present chapter. We will describe the main ideas in the proof of the general Theorem (1.3.5), and give a complete proof of the special result ( I .3.6).
Division systems and division spaces
Authors: Henstock Ralph 
Published: November 2023

Abstract
Generalized Riemann and variational integration theory connect a base space T of points of some kind, a space K of values, and three structures, (a) division systems and division spaces in T, (b) the algebra of K, (c) a topology or an order in K to define limits. As (a) is independent of (b) and (c) together, we can consider many combinations of the first structure with the other two.
Limits Under The Integral Sign, Functions Depending On A Parameter
Authors: Henstock Ralph 
Published: November 2023

Abstract
It is well known that the Lebesgue integral handles with ease many problems of the commutability of operations of integration and the taking of a limit, partially resolving C. Jordan’s question quoted by de la Vallee Poussin (1892a).
Give a rigorous theory of differentiation under the integral sign of definite integrals, with precise conditions which limit Leibnitz’s rule, principally for unbounded regions of integration or unbounded functions, and particularly many celebrated definite integrals.
Differentiation
Authors: Henstock Ralph 
Published: November 2023

Abstract
In Section 3.1 one of the limit processes is that of differentiation by a parameter, of an integral over a fixed elementary set E, (3.1.2) giving the one dimensional result. We can obtain necessary and sufficient conditions for
(3.1.2) to hold by translating into the present mathematical language the general Theorems 3.1.3 and 3.1.4.
Cartesian Products Of A Finite Number Of Division Systems (Spaces)
Authors: Henstock Ralph 
Published: November 2023

Abstract
Theorem 5.1.2 appeared first in Henstock (1961b), and the proof has been refined greatly since then. See also Kurzweil (1973a). Theorem 5.1.3 is due to T. W. Lee (1971/1972). In Lebesgue integration Theorem 5.1.2 in the symmetrical form is essentially due to Fubini (1907), and the unsymmetrical form is due to Cameron and Martin (1941) and Robbins (1948). See also Bosanquet (1930a).
Excesses and Coverings
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
There is a natural duality between packing triples on a set so that every pair appears at most λ times, and covering the pairs by triples so that every pair appears in at least λ triple. A covering by triplesof order vand index λ or CT( v, A),is a set V of velements, and a collection (multiset) of 3-elcment subsets of V(triplesas usual) for which every unordered pair of elements appears in at least Aof the triples.
Support Sizes
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
Triple systems are permitted to have repeated triples, triples that arc identical as 3-subsets. A theme throughout studies of triple systems is the goal to produce simpletriple systems having no repeated blocks. However, in some applications, the presence of repeated triples poses no difficulty. In fact, for certain applications in experimental design (Wynn, 1977; Foody and Hedayat, 1977), it is desirable to have many triples repeated.
Chromatic Number
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
Erdos and Hajnal (1966) and Lovász (1968, 1968a) were apparently the first to consider (weak) vertex-colourings of hypergraphs. Somewhat later, Berge (1973) formalized the notions of weak and strong chromatic numbers of hypergraphs. Both arc natural generalizations of the chromatic number of a graph: view a proper vertex-colouring of a graph either as one in which no edge is monochromatic (weak colouring)or as one in which the vertices incident upon a given edge all receive different colours (strong colouring).
Semigroup diagrams and word problems
Authors: Higgins Peter M 
Published: November 2023

Abstract
One of the few outstanding problems in algebraic semigroup theory that can be explained to a layman in a few minutes is the one-relator word problem. You are given two words, w1 andw2, together with another pair of words,u andv, all of whose letters come from some alphabetX. Is it possible to transform the word w1 into the wordw2, given thatu andv are treated as ‘equal’, meaning thatu andv can be interchanged, one for the other, as often as you please? The answer will of course depend on the given words. For example, let w1=ab2, w2 =ab2a2, with our ‘relation’ given byab= aba.
Around the spectrum
Authors: Helemskii A YA 
Published: November 2023

Abstract
The Latin word spectrum means something like “apparition”, “spirit” or “ghost”. But whereas the ghosts of kings and dukes, as a rule, inhabit ancient half-abandoned castles, the habitat of operator ghosts and, more generally, that of the elements of abstract algebras, is the complex plane, to which they were confined by Hilbert in his classic definition. In both cases the ghosts, if asked properly, can tell us many interesting things. Hamlet, the Prince of Denmark, in his day found this out in conversations with his father’s ghost. We hope that our reader will be no less impressed by the theorems on the representation of skew fields with continuous inverse (Theorem 1.17) and on the holomorphic functional calculus of a Banach algebra element (Theorem 2.15).
Polynormed algebras that are near to Banach algebras
Authors: Helemskii A YA 
Published: November 2023

Abstract
Historically, mathematicians became interested in general polynormed algebras (= locally convex algebras), as opposed to just Banach algebras, essentially for the same reasons which caused the theory of locally convex spaces to appear at a certain stage in the development of the theory of Banach spaces. The basic reason was, evidently, connected with applications: there had been an accumulation of significant “non-Banach” examples.
Topologists needed to work with the algebras C(Ω) when Ω was non-compact, complex analysists needed to work with algebras of holomorphic functions defined on domains in ℂn (specialists in operator theory needed to work with problems in functional calculus that also involved these same algebras), specialists in differential equations and harmonic analysis needed to work with various algebras of test functions and distributions. Another, this time intrinsic, stimulus was that studying structures that are more general than Banach algebras enables one to get a better understanding of the results of the “Banach” theory itself and to establish natural limitations to the scope of possible generalizations.
Introduction
Authors: Shawyer Bruce , Watson Bruce 
Published: November 2023

Abstract
Here we collect the chapter headings in the rest of the book and give a brief summary of the contents of each chapter.1 Historical overview A very brief survey of the subject of summability of series, ranging from the earliest attempts to sum convergent infinite series up to the first attempts to formalize the theory of summability of divergent series. For some of the material here we rely on a set of unpublished lecture notes of L. S. Bosanquet T[5].
Introduction
Authors: Yves Diers 
Published: November 2023

Abstract
The motivation for a study of the properties of categories of commutative algebras was the natural outcome of the publication of several papers developing techniques which mimic those of classical commutative algebral and algebraic geometry in areas such as the study of ordered algebra, lattice algebra, real algebra, differential algebra, and C∞ -algebra. For instance, G. W. Brumfiel’s Partially ordered rings and semi-algebraic geometry [4] develops a kind of ordered commutative algebra, the paper by K. Keimel entitled ‘The representation of lattice ordered groups and rings by sections in sheaves’ [28] develops a kind of lattice-ordered commutative algebra, the paper by M. E. Alonso and M. F. Roy entitled ‘Real strict localizations’ [1] develops real commutative algebra, the paper by W. K. Keigher entitled ‘Prime differential ideals in differential rings’ [27] develops differential commutative algebra and the paper by I. Moerdijk and G. E. Reyes ‘Rings of smooth functions and their localizations’ [32] develops C∞ commutative algebra. There are many other similar papers intent on extending the domain of interpretation of commutative algebra and transcending the particularity of ring structure. Thus the idea arose that the essence of commutative algebra is to be sought not in its calculus in commutative rings, but rather in the universal calculus in the category of commutative rings. It became natural to look for the universal properties in categories of commutative algebras which make commutative algebra work.
Schemes
Authors: Yves Diers 
Published: November 2023

Abstract
The localization process associates to any object a sheaf of local objects on its prime spectrum, while the globalization process rebuilds any object from the continuous family of its localized objects. By means of these processes, any object turns into a geometrical object: its affine scheme. By making these processes functorial, we obtain an equivalence between the category of affine schemes on a Zariski category A and the dual of A. Numerous properties of objects and morphisms in A have nice geometrical interpretations in the associated category of affine schemes. For example, a morphism in A is a regular epimorphism (or singular epimorphism etc.) if and only if its affine spectrum is a closed immersion (or open immersion etc.). Schemes on a Zariski category are built up by the classical method of gluing affine schemes together using modelled spaces and locally modelled spaces instead of ringed spaces and locally ringed spaces. Numerous classical properties and constructions of schemes are valid, finite limits and arbitrary coproducts of schemes exist, and the general gluing construction can be performed. Moreover, separated schemes are disjunctable; hence all disjunctors exist in the category of separated schemes and they are open immersions. In particular, affine schemes are disjunctable, and the open immersions into an affine scheme are precisely its singular subobjects.
Example: a two-point boundary value problem
Authors: Werschulz Arthur G 
Published: November 2023

Abstract
We use a simple example of a two-point boundary value problem (Lu)(x) = -u"(x) + u(x) = f(x) u'(O) = u'(l) = 0. for O &lt; X &lt; 1, to illustrate major issues and concepts of information-based complexity. First, we must describe our problem more carefully. That is, we specify the problem formulation, permissible information, and a model of computation.
General formulation
Authors: Werschulz Arthur G 
Published: November 2023

Abstract
Our main goal in this book is to find approximate solutions of differential and integral equations at minimal cost. Such approximations are given by optimal complexity algorithms. Instead of starting from scratch when finding optimal complexity algorithms for each new class of problems, it is more efficient to first recall general results from the theory of information based complexity, which we can then apply to each class of problems. As in the previous chapter, our key concepts are the problem formulation, the permissible information that may be used to solve the problem, and the model of computation that tells us which operations are permissible, and at what cost. We also introduce several settings, which allow us to measure the error and cost of algorithms. In particular, we are interested in the worst case setting (error and cost are measured at a hardest problem element), the average case setting (error and cost are defined on the average), and the probabilistic setting (error and cost in the worst case, but disregarding a set of problem elements of given measure).
Complexity in the asymptotic and randomized settings
Authors: Werschulz Arthur G 
Published: November 2023

Abstract
Recall that our emphasis so far has been on finding an algorithm that computes an ε-approximation over a set of problem elements, this computation being done at minimal cost. Moreover, we have only considered information and algorithms that are deterministic, not allowing for randomness. We now look at what happens if we measure what happens at a fixed problem element as the algorithms get successively better (the asymptotic setting) or if we allow nondeterministic algorithms (the randomized setting). We first look at the asymptotic setting. In this setting, we require that a sequence of algorithms approximate the solution at a particular problem element, and that this sequence converge as quickly as possible. The main result is that algorithms and information that are optimal for linear problems in the worst case setting are also optimal in the asymptotic setting.
Semigroup Kernels
Authors: Robinson Derek W 
Published: November 2023

Abstract
This chapter concentrates on general properties of the kernels, properties which are valid for all the kernels constructed from strongly elliptic operators. Then, in Chapter IV, we specialize to second-order operators with real coefficients. The semigroups generated by this latter class of operators have numerous special properties of positivity and dissipativity which can be exploited to refine the general properties.
Second-Order Operators
Authors: Robinson Derek W 
Published: November 2023

Abstract
The previous three chapters were devoted to the general theory of strongly elliptic operators but this chapter is much more specific. Its aim is the analysis of second-order operators with real coefficients, and the main em phasis is on unimodular groups. The previous theory of course applied to second-order operators but they also have many characteristic features not shared by their higher-order counterparts. For example, Theorem 111.5.1 established that the semigroup kernel corresponding to a strongly elliptic operator is positive if, and only if, the operator is of second-order with real coefficients. This positivity is just one aspect of the special properties of the second-order operators. Dissipativity is another.
On a problem of Wielandt and a question by Dembowski
Authors: Siemens J , Webb B 
Published: November 2023

Abstract
In Dembowski’s book (page 212) a similar question appears in a more restricted situation: If G is a group of automorphisms of a finite projective plane, acting primitively on lines, is G then necessarily primitive on points? The answer to this question is now known, due to the classification of projective planes with primitive collineation group by Kantor (1987).
The Gray Code Function
Authors: Clarke Francis 
Published: November 2023

Abstract
The Gray code of a positive integer m can be defined in terms of its binary expansion as follows. Shift m to the right, discarding the last digit, and then add the result to m without carrying (an exclusive or). Thus if we write ® for the exclusive or operation and ÷ for integer division, so that m÷2 is the right shift of m, then the Gray code of m is g(m) = m ® (m÷2). For each k, the function g permutes the integers between 0 and 2 k – 1. It has the property that the binary expansions of g(m) and (g(m+1) always differ in precisely one digit. This is true cyclically, in the sense that it holds for g(2 k – 1) = 2 k-1 and g(0) = 0. This provides a Hamiltonian circuit on the k-dimensional cube and is the basis of the construction of the diagram in Figure 1.
Out of the p-Adic into the Real
Authors: Everest G R 
Published: November 2023

Abstract
The theorem of this paper, when applied to the case of a profinite group, is well known, although a proof is not easily located. As a basic result on such groups, we feel that it deserves an explicit statement in the literature. We also give some applications to the continuous representation theory of a profinite group. We would like to thank the numerous people who made helpful comments on this result.
Packages for special occasions
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
Having given a detailed (but not comprehensive) overview of the SHEEP and CLASS! commands, we can now get down to applying these programs to specific problems in general relativity. The first problem we shall consider highlights the use of the MATTER package, written by Egils sviestins at Stockholm University.
Anholonomic frames
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
For most calculations in general relativity it is normally better to use anholonomic frames (tetrads), especially frames with respect to which the metric components are constants. Orthonormal frames are of particular importance since they represent the local frame of an observer whose four-velocity is the time like vector of the frame. Null frames also play a significant role and, to a lesser extent, half-null frames. Apart from their physical relevance, the components of the various tensors on these frames will usually be simpler and easier to handle than the corresponding ones with respect to the coordinate frames, since the “coordinate junk” has been eliminated.
Computation of the connection and the curvature
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
One of the earliest general relativity calculations performed with the aid of computer algebra was the computation of the Christoffel symbols and the components of the curvature tensor, Weyl tensor, Ricci tensor and Einstein tensor from the components of a given covariant metric tensor 9ij in some coordinate system. These elementary calculations provide a good illustration of the importance of the choice of algorithms for computer algebra programs in general relativity, a consideration that often seems to have been overlooked. Programs for performing the above calculations were usually based on standard formulas given in general relativity or tensor calculus textbooks. An examination of these formulas reveals that there is a variety of ways of computing the curvature tensor for example.
Computer algebra aided integration of field equations with the NP package
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
The usefulness of the Newman-Penrose (1962) (NP) formalism has been well-proven since its introduction, especially in the field of exact solution theory (see for example Kramer et al. 1981). In many instances, the manipulation of a system of first order differential equations in twenty-four complex indeterminates (i.e. the so-called spin coefficients and curvature quantities) is more conceptually straightforward than working in standard tensorial notation. However, due to the loss of compactness, expressions are often produced which are far too large to be dealt with effectively in hand calculations; this is often the case even when the desired result is simple and compact.
Introduction to reliability
Authors: Ansell J I , Phillips M J 
Published: November 2023

Abstract
The objective of this book is to describe elements of statistical data analysis which are useful in reliability studies, in a manner which gives the flavour of modern approaches. The chapter commences with a description of three examples of reliability studies taken from the authors’ practical experience of statistical data analysis. These studies, which cover a range of industrial contexts, are used later in Chapter 10 to illustrate the concepts presented throughout the book. It is important to define the objectives of a study and the objectives that can be achieved through the application of probability and statistical methods are outlined. The different forms of the data. encountered to achieve the objectives of a reliability study are illustrated in Section 1.4. These illustrations are made using examples from recently published papers in reliability journals and include right censored data, truncated data, accelerated failure data, and data from repairable systems. All these data must be collected and the proper planning of the way the, ae are collected is discussed. This leads to the consideration of databases, both the use of historic databases and the construction of new ones. However, before commencing any discussion of statistical methods a formal definition of reliability is required. The reliability of a system (or component) is defined as the probability that the system operates (performs a function under stated conditions) for a stated period of time. Usually the period of time is the initial interval of length t, which is denoted by [O, t)