bluuebunny/crossref_metadata_2025_split huggingface dataset:
Nearrings
Authors: Clay James R 
Published: November 2023

Abstract
Nearrings arise naturally in various ways, but most nearrings studied today arise as the endomorphisms of a group or cogroup object of a category. These nearrings are rings if the group object is also a cogroup object. During the first half of the twentieth century, nearfields were formalized and applications to sharply transitive groups and to foundations of geometry were utilized. Planar nearrings grew out of the geometric success of the planar nearfields and have found numerous applications to various branches of mathematics as well as to coding theory, cryptography, the design of statistical experiments, families of mutually orthogonal Latin squares and constructing planes with circles having radius and centre even though there is no metric involved. Even though nearrings may lack the extra symmetry of a ring, there is often a very sophisticated elegance in their structure. It has recently been observed that there is an abundance of symmetry in finite cirucular planar nearrings, which disappear if the nearring is a ring.
Order And Stability - A General Framework
Authors: Burrage Kevin 
Published: November 2023

Abstract
Chapter 2 attempted to give an overview and present a general structure in which many classes of differential equation methods can be studied. This third chapter continues the review of sequential methods and describes in more detail the properties of the numerical methods introduced in Chapter 2. This will be done from the framework of a very general class of methods known as multivalue methods. These properties can then easily be particularized to various classes of methods such as linear multistep and Runge-Kutta methods by understanding how these classes of methods fit into the multivalue framework. This is a very natural approach and one that has not really been attempted before, simply because not all the theoretical tools had been sufficiently developed. Recent developments now allow a systematic treatment of multivalue methods. However, it is important not to lose sight of the historical developments which have nearly always involved a movement from the particular (linear multistep and Runge-Kutta methods) to the general (multivalue methods).
Waveform Relaxation Techniques
Authors: Burrage Kevin 
Published: November 2023

Abstract
The standard application of initial value problem codes such as VODE (Brown and Hindmarsh (1989)) or STRIDE (Butcher et al. (1979)) can become inefficient for large systems of equations where different state variables have widely differing time rates. This is due to the fact that the same method and stepsize scheme is used in an attempt to represent accurately all components to a specified vector of tolerances. Various approaches have been proposed to overcome this drawback, based on partitioning (Hairer (1981)) and multirate (Skelboe (1989)) techniques. The idea here is to decompose the system into subsystems with similar time rates which can then be treated by possibly different methods and time-step strategies, more or less independently of one another. Of course the difficulty here is in knowing (automatically) how to couple the components and the nature of the coupling (perhaps variable) between the subsystems.
Implementation Of Waveform Algorithms
Authors: Burrage Kevin 
Published: November 2023

Abstract
The previous two chapters have focused on both waveform and discretized waveform algorithms. It has been shown that convergence results can be obtained for general splittings based on one-sided (L 1) and two-sided (L 2) Lipschitz constants, and that there is a remarkable similarity between these results for both the continuous and the discretized case.
Partial function theory, L′
Authors: Chapman Jonathan , Rowbottom Frederick 
Published: November 2023

Abstract
L’ is a conservative extension of the (canonical) local set theory L for a topos S (described in Section 2.12), to include symbols for partial functions in S. Beeson (1980/81, 1985, pp. 97—9), has (independently) considered a comparable first-order theory. Taking on board partial function symbols presents problems like those considered by Scott (1979), where he discusses a formulation of intuitionistic logic taking into account partially defined elements. L’ will be contrasted with this formulation. For more information on Scott’s approach, see Grayson (1978) where the set theory is developed in an abstract setting; Fourman and Scott (1979) for interpretation over a complete Heyting algebra; Fourman (1977) for the interpretation in an arbitrary topos. While Scott’s system has partial function symbols via descriptions, its interpretation in a topos is relatively complicated.
Wavelets and Spline Interpolation
Authors: Chui Charles K 
Published: November 2023

Abstract
Abstract. This is an introductory tutorial of wavelets from a spline theorist’s point of view. The concept of integral wavelet transform (IWT) for time-frequency localization is discussed, and a comparison of compactly supported orthonormal wavelets, spline wavelets, and biorthogonal wavelets is given. When spline wavelets are being used, interpolatory splines are readily available for orthogonal wavelet decompositions, yielding the IWT at dyadic locations without performing any integration. In addition, since the dual spline wavelets are used as wavelet window functions, linear-phase filtering is accomplished.
Preliminaries
Authors: Ellis Graham 
Published: November 2023

Abstract
There are many occurrences in mathematics of sets whose objects can, in some natural way, be added and multiplied. For instance we have the integers ℤ, the even integers 2ℤ, the rational numbers Q, the real numbers ℝ, and the complex numbers ℂ. Not all of the occurrences are as sets of numbers.
Error codes
Authors: Ellis Graham 
Published: November 2023

Abstract
In this chapter we study some applications of finite fields to coding theory. This is a theory concerned with the efficient transmission of digital information, and is essential for the proper functioning of electronic equipment such as computers and compact disk players. To explain the basic ideas behind the theory we begin by looking at a simple code used every-day by bookshops when ordering books. All modern books can be identified by their International Standard Book Number. For instance, the ISBN of A First Course In Abstract Algebra by J. B. Fraleigh is 0-201-16847-2.
Ruler and compass constructions
Authors: Ellis Graham 
Published: November 2023

Abstract
The last three chapters have been concerned primarily with finite fields. In this chapter we turn our attention to infinite fields and in particular to subfields of the real numbers ℝ. Our interest in such fields is derived from a construction problem in Euclidean geometry. Many geometric constructions can be performed using only a pencil, compass, and unmarked ruler. For instance, suppose we wish to bisect an angle θ, as in Fig. 5.1. This can be done, using our limited tools, as follows. Firstly use the compass to draw a segment of a circle cutting the sides of the angle at points A and B say. Then draw two circles of equal radius centred at A and B; these two circles intersect at points C and D, say, as in Fig. 5.2. The line CD bisects θ.
Pappus’ theorem and Desargues’ theorem
Authors: Ellis Graham 
Published: November 2023

Abstract
In Chapter 2 we saw that any field K gives rise to a projective plane PG(2, K). One very natural question to ask is: do all projective planes arise from a field in this way? In the present chapter we show that the answer is no. We do this by first proving that a classical theorem of Euclidean geometry, namely Pappus‘ theorem, holds in the projective plane PG(2, K) for all fields K, and then by exhibiting a projective plane in which Pappus‘ theorem does not hold. Another theorem of Euclidean geometry which may or may not hold in a given projective plane is Desargues‘ theorem. A famous result of projective geometry states that in any finite projective plane Pappus‘ theorem is implied by Desargues‘ theorem.
Do it with groups
Authors: Cofman Judita 
Published: November 2023

Abstract
The theory of groups is an important branch of modern algebra, with various applications in mathematics and in physics. The notion of a group is based on that of a binary operation on a set. A binary operation on a non-empty set S is a rule which associates to every ordered pair (a, b) of (not necessarily distinct) elements a, b of S a unique element of S.
Chapter IX
Authors: Cofman Judita 
Published: November 2023

Abstract
Let n be an arbitrary integer. ⊕ is a binary operation on ℤn = {O, 1, ... , n - 1} because i ⊕ j ∈ ℤn for any i,j ∈ ℤn, It is easy to verify that ℤn (⊕) satisfies all group axioms:.
MFE In 1-D: Second-Order Equations
Authors: Baines M J 
Published: November 2023

Abstract
The MFE method has most often been used on second-order PDE’s involving two space derivatives of u. Indeed, the method was introduced specifically for problems involving convection and diffusion with steep moving fronts in the solution (see Chapter 2, Sections 2.7 and 2.8) (Miller and Miller 1981, Gelinas et al. 1981, Djomehri 1983). Although in diffusion problems we cannot expect any link with the approximate method of characteristics of Chapter 4 the MFE method is still a method of trajectories and we need to understand the properties of the nodal speeds. To do this we make a special study of the linear heat equation
Diophantine Equations
Authors: Rose H E 
Published: November 2023

Abstract
Diophantus worked in Alexandria probably in the middle of the third century AD (some authors suggest early in the second century AD). His six or seven surviving books show that he had a highly developed understanding of many number theoretical problems. These mainly involved the solution of equations in positive integers or positive rational numbers although his methods of working suggest that he was aware of negative numbers. He was one of the first to introduce symbolic notation into mathematics; for example.
Nonlinear acoustic surface waves in crystals
Authors: Maugin Gerard A 
Published: November 2023

Abstract
In general, surface waves represent time-varying and spatially nonuniform perturbations. The spatial variation in the field amplitude is markedly confined to the vicinity of the limiting surface of a body and practically vanishes outside this relatively narrow zone. They are examples of the so-called inhomogeneous waves. The treatment of surface waves is one of the simplest-but not necessarily easy-dynamic problems concerning a body with finite extension along one of its dimensions. The boundary conditions must be considered such that, at least in principle, any field theory and the resulting continuous field equations exhibit some surface wave phenomenon.
The second-order deformation of an incompressible isotropic slab under torsion
Authors: Lindsay K A 
Published: November 2023

Abstract
The theory of elasticity has been traditionally divided into problems associated with linear theory and those involving the general theory. The method of successive approximations seeks to bridge this division by expressing general solutions in terms of the ‘small‘ parameter of linear theory although in practice advancement beyond linear theory is often only technically feasible if the linear solution is either mathematically elementary or the problem enjoys a strong degree of symmetry. Questions concerning existence and uniqueness of solutions have been addressed by Signorini (1) and Stoppelli (2, 3) under suitable conditions of differentiability. For many problems in linear elasticity, the method of successive approximations has no relevance since linear theory encapsulates all the essential features of a realistic solution. However, there are some problems involving either torsion or shear in which linear theory appears inadequate in the sense that some component of the linear solution is zero whereas reality suggests otherwise. In such problems, the theory of successive approximations has a valuable role to play in the determination of such components of the leading solution. Second-or higher-order theories must be derived directly from general theory: they cannot be derived from linear theory as it is already approximate. In particular, higher-order theories differentiate between reference and deformed coordinates (unlike linear elasticity) and for this reason it is convenient to work with reference coordinates. One consequence of this decision is that the appropriate stress tensor may no longer be symmetric although in this application, symmetry does prevail. Further details are available in Green and Adkins (4)
Finite-amplitude waves in deformed Mooney-Rivlin materials
Authors: Boulanger Ph. , Hayes M 
Published: November 2023

Abstract
In a previous paper (3), Currie and Hayes showed that two linearly polarized finite-amplitude shear waves, polarized in directions orthogonal to each other and to the direction of propagation n, may propagate along any direction in a Mooney-Rivlin material which is maintained in a state of arbitrary static finite homogeneous deformation.
Closure of an arc crack in an isotropic homogeneous material due to uniaxial loading
Authors: Chao† R , Laws N 
Published: November 2023

Abstract
In this paper we consider the problem of an arc crack in an infinite isotropic elastic solid under uniaxial loading at infinity. The major analysis is undertaken with the help of complex-variable methods. A point of particular concern is that existing analyses do not properly account for partial crack closure due to load orientation. A proper solution is given for partially closed cracks. This solution is compared with the standard open-crack solution of Muskhelishvili (1). For most purposes the classical solution is adequate†but there are significant differences.
Deformations of an elastic, internally constrained material part 2: nonhomogeneous deformations
Authors: Beatty Millard F , Hayes Michael A 
Published: November 2023

Abstract
This is a sequel to part 1 in which a theory describing the nonlinear mechanical response of a class of homogeneous, isotropic elastic materials subject to finite deformations and characterized by Bell’s constraint tr V = 3 was presented. Although Bell discovered this constraint within the context of experimental work in finite-strain plasticity, in part 1 it was examined for homogeneous deformations in the context of finite elasticity. Here the emphasis is on non-homogeneous, finite deformations for isotropic, homogeneous, elastic Bell-constrained bodies. Six families of solutions are presented. They include the bending and stretching of a rectangular block; the straightening of a cylindrical sector into a rectangular block; the bending with axial stretch of one circular cylindrical wedge into a similar wedge; the inflation, bending, extension, and azimuthal shearing of an annular wedge; the radial deformation of a spherical shell and membrane—including inflation, compression, and eversion; and, finally, the equibiaxial stretch and sinusoidal shear of a block.
Representation Groups for the Symmetric Group
Authors: Hoffman P N , Humphreys J F 
Published: November 2023

Abstract
In this chapter, we shall determine the Schur multiplier of the symmetric group S n, and give presentations for it representation groups. To find an upper bound for M(S n), we shall present a portion of Schur’s method to calculate M(G) starting from a presentation of G. For this, two basic results in group theory, Theorems 2.1 and 2.3 below, are quoted without proof. Alternatively, there is a direct approach to this upper bound which uses the twisted group algebra and the existence of a projective representation for any given cocycle; for a treatment, see Jozefiak (1989), pp. 198-200. The first quoted result consists of some consequences of the classification theorem for finitely generated abelian groups.
Representations of Objects in 𝒢
Authors: Hoffman P N , Humphreys J F 
Published: November 2023

Abstract
In this chapter, we shall discuss the representation theory of an object (G,z, σ) in 𝒢. To avoid unnecessary repetition, the word “representation” will be used to mean “linear representation” as defined in Chapter 1. We shall restrict attention to representations over the complex numbers. Character theory can then be used in many proofs. This is a suitable point, therefore, for a short digression recalling the main results of character theory. The following six paragraphs, labelled (Di), will each conclude with a statement of a property of characters labelled (Ci). The reader should consult Huppert (1967) for proofs of the non-obvious facts below. The references given in parentheses are to that book.
Reduction, Branching and Degree Formulae
Authors: Hoffman P N , Humphreys J F 
Published: November 2023

Abstract
The relationship between combinatorics and linear representations of the symmetric group has proved to be of mutual benefit to both. Much recent progress on the projective representation theory of the symmetric group depends on a combinatorial framework which allows one to develop a similar relationship. The main idea is that of a shifted diagram, which we introduce next. The reduction formula and branching rule are then interpreted combinatorially using diagrams. Finally, the degrees of the irreducible projective representations of S n are calculated and related to the idea of a standard shifted tableau.
Rounding errors in algebraic processes¬ in level-index arithmetic
Authors: Olver F W J 
Published: November 2023

Abstract
The level-index number system represents numbers in a computer by their repeated logarithms. Its chief advantage is closure in finite¬ precision arithmetic, thereby eradicating the problems of overflow and underflow. This talk indicates how a Wilkinson-type running error analysis can be carried out in the new system.
Oscillations Of Systems Of Delay Equations
Authors: Györi I , Ladas G 
Published: November 2023

Abstract
This chapter deals with the oscillation of all solutions of systems of delay differential equations. Section 5.1 gives a powerful necessary and sufficient condition for the oscillation of all solutions of a linear autonomous system. Section 5.2 utilizes this result to obtain explicit conditions for the oscillation of linear autonomous equations. Section 5.3 deals with the oscillation of linear, non-autonomous equations. Section 5.4 provides sufficient conditions for the oscillation of all solutions of a non-linear system of delay logistic equations.
Miscellaneous Topics
Authors: Györi I , Ladas G 
Published: November 2023

Abstract
Sections 12.1 and 12.2 contain some simple but interesting results on the non-existence of slowly oscillating periodic solutions. Section 12.3 deals with necessary and sufficient conditions for all solutions of a neutral equation with periodic coefficients to oscillate. In Section 12.4 we obtain a result on the growth of rapidly oscillating solutions of some linear equations.
The mechanization of computation at the Nautical Almanac Office
Authors: Croarken Mary 
Published: November 2023

Abstract
During the 1910s and 1920s scientists were, in general, slow to adopt desk calculating machines for their work. Small computing laboratories had been set up by Pearson and Whittaker but in most other institutions routine computations were carried out using logarithmic tables. As outlined in the previous chapter the British Nautical Almanac Office (NAO) at Greenwich was no exception.
The alleged virtues
Authors: Hogger Christopher John 
Published: November 2023

Abstract
Many arguments have been advanced in favour of logic programming. The principal argument is that it enables one to express knowledge explicitly in a machine-independent manner, making programs more compact, flexible and intelligible. The thesis that the representation and derivation of knowledge is what computing is all about is not, at present, conceded by the majority of programmers, and even those who do concede it do not necessarily see logic programming as the natural way of realizing it. However this may be, the main strands of the thesis are as follows.
Logic programming can be regarded as the corner-stone of knowledge based programming. The expressiveness of logic enables one to formulate (encode) assumptions about the problem domain in a direct, implementation¬ independent manner. Conversely, one can readily decode such a formulation in order to recover its underlying assumptions. Furthermore, the conclusions computed from the formulation are exactly its logical consequences, and so are directly relatable to the assumptions.
Clausal-form logic
Authors: Hogger Christopher John 
Published: November 2023

Abstract
Clausal form is one of the many normal forms of first-order logic and is the sublanguage in which pure logic programs are conventionally written. The common feature of all normal forms is that they make the structure of sentences more regular. Whatever sentences we might initially construct in arbitrary first order logic in the course of formulating some problem, those sentences can always be converted into new clausal-form sentences without any loss of expressiveness. We define a sentence to be in clausal form if and only if it takes the form &lt;universal-prefix&gt;&lt; matrix&gt; where the universal-prefix is a string of universal quantifiers, the matrix is a quantifier-free conjunction of formulas and each of those formulas is a disjunction of one or more literals.
Herbrand procedures
Authors: Hogger Christopher John 
Published: November 2023

Abstract
In the early days of automated theorem proving much use was made of the principles discussed in the last Theme: replace the given clause-set by its ground instantiation; then seek a contradiction. The formal justification for this approach relies upon a profound and highly important result in proof theory known as Herbrand’s Theorem. There are many ways of expressing and specializing this theorem, but the specialization of it which follows is perhaps the one best-suited to our present context:
Applying and composing substitutions
Authors: Hogger Christopher John 
Published: November 2023

Abstract
When resolution is used for clausal-form predicate logic, the task of constructing a resolvent generally requires substituting certain terms for variables in the literals which the resolvent inherits from its parents. Later on we shall say exactly how this is done, but in the meantime it will be useful first to make precise what we mean by ‘substitutions’ and then to consider some of their properties.
Resolution versus Herbrand procedures
Authors: Hogger Christopher John 
Published: November 2023

Abstract
It is instructive to see what would be involved in using a Herbrand procedure in order to solve the ‘grandparent’ problem presented in the last Theme:
According to the Herbrand Theorem, the ground instantiation is contradictory (inconsistent) if and only if it includes some finite, contradictory subset. One such subset consists of the four clauses shown above in bold face. It is trivial to derive
The occur-check
Authors: Hogger Christopher John 
Published: November 2023

Abstract
The unification algorithm presented in the previous Theme includes in its comparison cycle the following test: else if either of e1 and e2 is a variable which occurs strictly within the other then issue “failure” and tem1inate This test is called the occur-check and its purpose is to disallow self-referential bindings such as X/f(X). Here is a simple example in which that binding would arise in the unifier if the occur-check were to be omitted:
The binding X/f(X) effectively assigns to X the infinite term f(f(L)), whereas all expressions in our language are presumed to be syntactically finite. In particular, although the Herbrand domain might well be an infinite set of terms, the terms themselves are presumed to be finite.