bluuebunny/crossref_metadata_2025_split huggingface dataset:
Basic stock of examples
Authors: Helemskii A YA 
Published: November 2023

Abstract
Our collection of “on call” examples of algebras, given in Chap.I § 2.2, is no longer enough if we want to move on. In this chapter we get to know some new specific algebras and we also study more thoroughly the examples we already know. The algebras considered here, in much the same way as our subject itself, belong to neighbouring branches of analysis the theory of functions, operator theory, harmonic analysis; there are some questions in topology and in the mathematics of quantum mechanics that are also related to this same boundary layer. The application of the general theory of algebras of analysis to these disciplines is through just such algebras as these: to obtain information about a topological group it is necessary to involve its group algebra; to examine an “individual” operator, it is useful to include it in some operator algebra or other, and so on. On the other hand the material in this chapter is needed to get an informal grasp of the constructions and results of the general theory, to which we shall return in the next few chapters: there we shall have far more opportunity to show how all this works and what it turns out to be in specific situations
Stratification theorems
Authors: Evens Leonard 
Published: November 2023

Abstract
Let k be a field of characteristic p, G a finite group, and E an elementary abelian p-subgroup of G. We shall identify Eˆ= Hom(E, F p) with Im{H 1(E, F p) → H 1(E, k)} for p = 2 or Im{H 2(E, Z) → H 2(E, k)} for p odd. (For p odd, Eˆ may also be thought of as the image of the composite map [ineq] Similarly, we shall identify E* = Hom(E, k) with the k-subspace of H(E) generated by Eˆ: (This is a subspace of degree 1 for p = 2 and degree 2 for p &gt; 2.) For p = 2, H(E) = H*(E, k) ≅ S(E*) (the symmetric algebra on E*). For p &gt; 2, H(E) = S(E*) ⊕ J where J is the nilpotent ideal generated by H 1(E, k)2 ⊂ H 2(E, k); in fact, J = rad H(E).
Summability methods in general
Authors: Shawyer Bruce , Watson Bruce 
Published: November 2023

Abstract
A summability method for a series [ineq] is simply any scheme for as- signing to it a unique number which we then call the ‘sum’ of the series or the ‘limit’ of the associated sequence of partial sums {sn}, In practice, working with a method can usually be reduced to consideration of one of the following transformations.
Applications of Borel’s methods
Authors: Shawyer Bruce , Watson Bruce 
Published: November 2023

Abstract
The bibliography of articles that can be classed as applications of Borel’s methods has about 70 entries. Of these, only one predates 1970. That is the article by Cunningham in 1905 A[18] entitled An extension of Borel’s exponential method of summation of divergent series applied to linear differential equations. Here, Cunningham’s object was to make more precise the connection between Laplace’s integral solutions of linear differential equations with rational coefficients and of rank unity in the neighbourhood of z = ∞, and the power series solution with an exponential factor. He noted that Poincaré had developed asymptotic expansions of the integrals, and had found them useful for numerical calculations. However, these asymptotic expansions were of little value in that they did not represent unique analytic functions.
Zariski Categories
Authors: Yves Diers 
Published: November 2023

Abstract
We use the axiomatic method, which deals with categories equipped with a specified structure rather than categories described concretely by their objects and their morphisms. The so-called Zariski categories have such a structure, and we claim here that it is the basic structure of categories of commutative algebras. We shall prove the validity of this claim by developing, in an arbitrary Zariski category, elementary commutative algebra and algebraic geometry. The structure of Zariski categories relies on the notion of codisjunctors. This universal construction is introduced in order to describe the calculus of fractions and is used extensively. In Zariski categories, the congruences, also called effective equivalence relations, play the important role of ideals. A kind of De Morgan law for codisjunctors of congruences is assumed. Among the basic properties of Zariski categories are the flatness properties. We use the following definition: a morphism f: A → B is flat in A if the pushout functor A/ A→ B/A along f preserves monomorphisms. Codisjunctors are assumed to be flat morphisms.
Review of stability theory
Authors: Du Plessis Andrew , Wall Terry 
Published: November 2023

Abstract
The earliest results in the theory of stability of smooth mappings are due to Whitney. The case of maps ℝn →with ℝp p 2 ≥2n -1 is elucidated in Whitney (1943) and the case p = n = 2 in Whitney (1955). In Whitney (1958) these and other results are summarized, possible generalizations discussed, and the general conjecture formulated that a generic map has all its germs c ∞=-stable; also, this is placed in a function space setting.
Other problems in the worst case setting
Authors: Werschulz Arthur G 
Published: November 2023

Abstract
In the previous chapter, we applied the techniques of information-based complexity to find algorithms that are optimal in the worst case setting for elliptic partial differential equations (PDE). In this chapter, we apply these techniques to find algorithms that are (worst case) optimal for several other important classes of problems. The first section deals with elliptic systems of PDE. The second section deals with well-posed integral equations (IE), namely the Fredholm problem of the second kind. The third section is a general treatment of linear ill-posed problems, with special emphasis on the Fredholm IE of the first kind. Finally, the fourth section deals with ordinary differential equations (ODE).
Functional Spaces in Viscoplasticity
Authors: Ionescu Ioan R , Sofonea Mircea 
Published: November 2023

Abstract
The purpose of this chapter is to provide some fundamental results on functional spaces, as used in viscoplasticity, for later reference. These spaces are various spaces of distributions associated with the deformation operator and the divergence operator, or spaces of vector distributions. To restrict the chapter to a reasonable length most of the basic theorems are stated without proofs. However, for the convenience of the reader some useful properties of the spaces related to the divergence operator are presented here with proofs (Section 2.3). For other results which can be easily found in text books or monographs adequate references are given. The notations stated in this chapter are used all over the book.
Introduction
Authors: Robinson Derek W 
Published: November 2023

Abstract
The purpose of this book is to develop the basic theory of elliptic opera tors on Lie groups and thereby extend the conventional theory of parabolic evolution equations to a natural non-commutative context. The use of a Lie group as the underlying manifold introduces many complications not encountered on Euclidean space. The usual partial differential operators are replaced by representatives of the corresponding Lie algebra and this leads to a variety of difficulties even in the description of operators with constant coefficients. The Lie group theory of strongly elliptic operators with constant coefficients is comparable in complexity to the Euclidean theory with variable coefficients. Locally the two theories are in fact equiv alent and this is the key to the construction of the Lie group theory. The basic existence, regularity, and boundedness, properties are established by means of local Euclidean estimates.
On totally irregular simple collineation groups
Authors: Gorn;alves C Y Hoand A 
Published: November 2023

Abstract
Classifications of Desarguesian planes by doubly transitive collineation groups (Ostrom-Wagner) or by the property that either each point is a centre or each line is an axis of a perspectivity (Cofman Piper) are examples of totally irregular collineation groups with few orbits. Cyclic planes (in general planar difference sets) also admit totally irregular collineation groups, by using Hall’s multiplier theorem (Hughes and Piper (1973)).
Stabilisers of hyperovals in PG(2, 32)
Authors: O’Keefe Christine M , Penttila Tim , Praeger Cheryl E 
Published: November 2023

Abstract
We shall be concerned with the collineation stabiliser G and the homography stabiliser H of a hyperoval ℋ in PG(2, q), that is, the stabilisers of 5/ in the collineation group PTL(3, q) and homography group PGL(3, q) of PG(2, q) respectively.
Applications of the p-Adic Subspace Theorem
Authors: Everest G R 
Published: November 2023

Abstract
Given an integer m⩾2 and rational numbers A1,…, Am, u1,…, Um not all zero, we obtain a rational sequence using the linear recurrence relation, [eq] Sequences of this kind have been studied extensively and the theory has a long and colourful history. One of the earliest and best known is the Fibonacci sequence. This begins u1 = u2 = 1, and continues via the recurrence relation A slightly more exotic example occurs with the problem of the man who walks along an infinite ladder.
Arithmetical homomorphisms
Authors: Klingen Norbert 
Published: November 2023

Abstract
From Chapter III, theorem (1.4) we know that arithmetically equivalent fields have lots of invariants in common; in particular their zeta functions agree. From this it followed that the products hR of class number and regulator coincide in arithmetically equivalent fields (III.1.4 j)), but the relation between the class numbers themselves remained open. So one important subject in this chapter is the investigation of class numbers.
Moduli Space For Invariant Self-Dual Virtual Connections
Authors: Petrie Ted , Randall John 
Published: November 2023

Abstract
Let G be a compact Lie group. Previously (Section 2.8) the term G-vector bundle meant a vector bundle with group G as the structure group. It has another accepted meaning (used throughout the remainder of this text) when G acts on the base space of the vector bundle. In this case there are two groups to deal with, G and the structure group of the bundle. We shall deal exclusively with the structure groups SO(2) and SO(3). In this context, a G-vector bundle is a vector bundle (of dimension 2 or 3) with G acting to preserve the inner product and orientation, commuting with base projection, and mapping fibres directly to fibres (see [1]).
Using CLASSI to classify metrics
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
The CLASSI package (Åman 1987) is introduced in this chapter by an account of the theoretical basis of the classification procedures which it embodies. Then the direct use of the classification programs is described, followed by further details of extra facilities such as the handling of complex quantities. The form of the results and the future directions of work in this area are considered. Finally a list of all CLASS! modules, including those not directly related to the classification algorithms, is given.
Introduction
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
The lecture notes that follow are aimed at those who have already done a first course in general relativity and have acquired an elementary knowledge of programming in the algebraic mode of REDUCE 3.3 (Hearn 1987). The intention is not to present the reader with a software package that will function as a sort of black box which processes input and churns out output. The reader is rather invited to work through the details of the program given here and then to go and write his/her own program, at least at the level of the algebraic mode which is all that will concern us here. In several places in the text, reference is made to a library of files called GRLIB. These are a set of straightforward REDUCE programs for computing the more commonly used geometrical objects in general relativity.
Calculations on coordinate bases
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
The golden rule in performing general relativistic calculations is to use, whenever possible, anholonomic frames for which the metric components are constants. The reason for this is that the expressions one obtains for the components of the various tensors that arise are normally simpler in an anholonomic frame. However sometimes it is necessary or convenient to calculate with a straightforward holonomic or coordinate frame and it is to this type of computation that we first turn our attention in this section. A typical requirement would be, for instance, to compute the Riemann, Ricci or Einstein tensors for a given space time metric, or to look for the Killing vector fields of the space-time. If there are also electromagnetic fields present the Einstein-Maxwell equations will have to be computed or perhaps one may wish to apply the Rainich (1925) conditions.
Introduction: overview of MAPLE applications to general relativity
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
For the past thirty years researchers have been using computer algebra systems to perform some of the lengthy and tedious computations involved in research in general relativity. The pioneers of the subject Fletcher et al. (1967), d’lnverno (1969), Barton and Fitch (1972), and Pavelle (1975) mainly used the systems to compute the Christoffel symbols and the components of the curvature tensor, Weyl tensor, Ricci tensor and the Einstein tensor from the components of a given space time metric in the natural basis. The main purpose was to verify that the metric actually satisfies Einstein’s field equations. A further early application was the determination of the Petrov type for all the Harrison’s (1959) solutions by d’Inverno and Russell-Clark (1971). Programs were also written by Ernst (1968) and Harrison (1969) for the computation of the connection coefficients and curvature components, corresponding to a given rigid null or orthonormal frame using Cartan’s structural equations.
The Petrov classification of the Weyl tensor
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
The determination of the Petrov type of the Weyl tensor for a given space-time metric is important in exact solution theory, in particular in the equivalence problem where it forms an important part in the algorithm for determining whether two space-time metrics are locally isometric. The determination of the Segre type of the Ricci tensor (not discussed here) is also an important part of the algorithm. It should be noted that most Ricci tensor classification algorithms (see Joly and MacCallum 1990) require the determination of the Petrov type of the Plebanski tensor. Thus the determination of the Petrov type of tensor with Weyl tensor symmetries is seen to be particularly fundamental especially in the context of the equivalence problem.
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Analysis of lifetimes with covariates
Authors: Ansell J I , Phillips M J 
Published: November 2023

Abstract
In many reliability studies as well as the information on the component’s (or system’s) lifetime there is also available other information about the component (system) or its environment. Typically there may be information on the design of the component or the wear the component has suffered. Hence instead of just recording a set of lifetimes there may be associated with each lifetime other variables. These variables are known as covariates. (The term is often used to cover factors, see Section 3.2, as well as measurable variables.) This extra information should yield more understanding about the performance of the component or system. The more information used the better the understanding, hopefully. Recently there has been considerable interest in the analysis of such data sets in the reliability and related literature. Part of the interest may be due to the availability of the proportional hazards model, suggested by Cox (1972, 1975). This has focused interest in relating both variables and factors to components’ lifetimes. However, the proportional hazards model is not the only model available and a number of approaches predate it, such as the use of the accelerated failure time model (Nelson, 1993) and the Weibull regression model (Smith, 1991). All the models and techniques discussed in this chapter are regression-type techniques.
Practical aspects of reliability data analysis
Authors: Ansell J I , Phillips M J 
Published: November 2023

Abstract
Previous chapters have explored models and techniques which provide the basis for reliability data analysis. In this chapter the practical aspects are discussed. Before starting on such a discussion, though, it is worth re marking that many reliability analyses do not require the use of statistical techniques. Even for those which do, the use is often very limited and routine. Sometimes their use may even be misguided, see O’Connor (1991). The statistical analysis will often not play the central role in the full analysis and will mainly be used to support the conclusions about the main objective. However, the use of statistical analysis can provide considerable information to enable analysts to make decisions. Often the value of this analysis is overlooked and the full power of the statistical tools available underutilized. One reason is that in the past reliability engineers have seen sophisticated techniques applied to their data with relatively little advantage to them. There has been a concentration on the technique rather than the underlying problem. When statistical methods are used the approach should be problem-led rather than technique-led. This means that the objective of the analysis should always be to the fore in the study. The objective is the key to the analysis. The way to achieve the objective is through a strategy. The first section of this chapter will there- fore concentrate on objectives and strategies. The strategy offered in this chapter is not prescriptive; hence unlike other authors a flowchart is not provided for the possible stages in the analysis. Obviously different objectives will have specific strategies developed for them.
Topological homology
Authors: Eschmeier Jörg , Putinar Mihai 
Published: November 2023

Abstract
The homology of topological algebras and of topological modules over topological algebras has been developed relatively late and rather at a tangent to the main directions of research in homological algebra. However, this gap has been considerably reduced in recent times, when the interest in the Hochschild cohomology of operator algebras has increased, partly due to the elaboration of cyclic cohomology and of non-commutative geometries.
The first limit problem in MPT
Authors: Grenander Ulf 
Published: November 2023

Abstract
In this and the following two chapters on metric pattern theory (MPT) we shall study probabilistic limit theorems on regular structures. They are related to the limit theorems or standard algebraic structures such as semigroups, groups, vector spaces and algebras (Notes 9.A) but differ from them in that the algebraic properties of regular structures, as studied in Part I, are quite different. Instead of ‘addition’, ‘multiplication’ by scalars, or whatever the unary and binary operations are called in standard situations, we now have the laws of local interaction, action of similarity transformations and coupling connectors.
Continuum-based models
Authors: Grenander Ulf 
Published: November 2023

Abstract
In this first section we shall study picture ensembles consisting of human hands captured by a visible light digital camera under various degrees of optical degradation. Hand shapes are very characteristic but also exhibit a lot of variability, so that inferences take place in a high-or infinite-dimensional parameter space – abstract inference.We aim for both intrinsic image understanding (see Chapter 13) in the form of image restoration and objective landmark determination and for extrinsic understanding, here meaning abnormality detection.
Hamilton-Jacobi Theory
Authors: Woodhouse N M J 
Published: November 2023

Abstract
Eqn (4.1.1) is the Hamilton-Jacobi equation (more accurately, it is the time-independent form of the Hamilton-Jacobi equation). Although it was originally introduced as a device for obtaining analytic solutions to mechanical problems, it is, in fact, of very little use in that context since the only generally applicable method for solving such first-order partial differential equations involves the reverse construction and depends, as a first step, on the integration of Hamilton’s equations (for example, see Courant and Hilbert 1962). Apart from this the only other practical method is to separate the variables, and this can only be done for a very special class of Hamiltonians.
Classical Fields
Authors: Woodhouse N M J 
Published: November 2023

Abstract
At a formal level, much of the symplectic geometry that underlies the Hamiltonian formulation of classical mechanics extends to field theories, and can be used as the starting point for quantization. In this chapter, we shall look at relativistic field theories, and at the connection between the Lagrangian density and the symplectic structure on the space of solutions.1 To prepare the ground for quantization, we shall look at how the splitting of the fields into their positive and negative frequency parts can be interpreted as a complex polarization in the classical phase space.
Discrete Wavelet Transform
Authors: Freeden W , Gervens T , Schreiner M 
Published: November 2023

Abstract
In the wavelet reconstruction formulae studied so far, the discretization of the corresponding wavelet transforms must be made directly by computing the relevant integrals for all the necessary values of rotation parameters. In Euclidean theory, however, a multiresolution analysis for performing discrete wavelet analysis and synthesis is known, which is completely recursive and, therefore, ideal for computations. This form of multiresolution analysis is certainly one of the reasons why the wavelet theory has gained such widespread popularity. In what follows we make the attempt to generalize as much as possible from this multiresolution procedure to the spherical context.
Open problems
Authors: Baaz Matthias 
Published: November 2023

Abstract
1 Bounded Arithmetic
The study of theories of bounded arithmetic was largely begun by work of R. Parikh [51] and the ground-breaking work of J. Paris and A. Wilkie [54, 73, 58, 57, 72, 25]. Especially with the work of S. Buss [10], proof theoretic problems of weak theories of bounded arithmetic were seen to be connected with problems of computational complexity theory.
0 Introduction
Authors: Gurtin Morton E 
Published: November 2023

Abstract
In this treatise I discuss the dynamics of two-phase systems within the framework of modern continuum thermodynamics, assuming throughout a sharp interface separating bulk phases. I consider only non-deformable continua, neglecting mass transport, and to avoid the geometrical complications associated with the motion of surfaces in ℝ, I restrict attention to evolution in the plane. I consider two general theories: a purely mechanical theory that leads to a generalization of the classical curve-shortening equation