bluuebunny/crossref_metadata_2025_split huggingface dataset:
Eigenvalue problems with indefinite weights and reaction-diffusion models in population dynamics
Authors: Cosner Chris 
Published: November 2023

Abstract
The object of this article is to illustrate how the theory of linear elliptic eigenvalue problems with indefinite weight functions can be used to analyze reaction-diffusion models in mathematical ecology and population genetics. It is not intended to be a comprehensive survey, but rather an introduction to some of the principal ideas and methods. Reaction-diffusion equations arise in ecological modelling when nonlinear dynamics describing the growth or decline of a population are combined with a diffusion process describing the spatial dispersal of that population. Solutions to the reaction-diffusion models typically represent population densities, or in related problems of population genetics, the distribution of certain alleles within a population.
Network Reliability and Algebraic Structures
Authors: Shier Douglas R 
Published: November 2023

Abstract
Networks of computer systems, distribution systems, and telecommunication systems play an increasingly important role in our lives. This book is concerned with analysing and assessing the reliability of such networks whose components are prone to possible failure. The field of network reliability has expanded rapidly to provide various mathematical models and computational procedures. In this book the emphasis is on identifying a number of algebraic structures that serve to unify the study of networks, and to suggest new techniques and procedures for analysing the probabilistic behaviour of networks. Consequently, amongst the topics covered are pseudopolynomial algorithms, lattice structures, spaces of polynomials, reliability covering problems, and stochastic path problems. The result is a book which will be an invaluable account of the subject for all mathematicians, computer scientists, reliability engineers, and operations researchers whose work touches on network reliability.
Overview of network reliability
Authors: Shier Douglas R 
Published: November 2023

Abstract
The broad theme of reliability theory is the study of the overall performance of a system comprising failure-prone elements. That is, the components of the system are not perfect in operation, but their failure is assumed to be governed by certain probabilistic laws. It is thus of interest to describe the stochastic behavior of the system in terms of the given stochastic characteristics of its components. Not only are the reliabilities of individual components of importance, but the manner in which they are assembled can have a significant influence on the overall performance of the system. For example, as noted over thirty years ago by Moore and Shannon (1956), it is possible to obtain a reliable system by properly configuring unreliable components through the use of redundancy.
An algebraic formulation of network reliability problems
Authors: Shier Douglas R 
Published: November 2023

Abstract
This chapter presents an algebraic approach for studying network reliability problems, concentrating, in particular, on the calculation of R(G). Instead of working with numerical quantities (probabilities), the emphasis is on algebraic objects (polynomials) and their appropriate transformation. This algebraic perspective not only provides a unifying theoretical framework for studying network reliability problems, but also suggests a number of new algorithmic approaches. We will see, in particular, that several standard linear algebraic techniques can be readily adapted to solve network reliability problems. In this case, however, symbolic manipulations rather than numerical calculations will be carried out. The symbolic form of the resulting answer is especially useful in conducting sensitivity analyses, since the change in overall reliability with changes in component reliabilities can be directly ascertained from this form.
Planar nearrings
Authors: Clay James R 
Published: November 2023

Abstract
In the summer of 1967, the author hoped to discover some geometric interpretations for nearrings. Since planar nearfields had already proven useful to geometry, he hoped that planarity could be extended to nearrings which were not nearfields. After the author developed a suitable concept for planarity for nearrings, Michael Anshel constructed numerous examples in which the algebraic structure of planar nearrings had nice geometric interpretations. Some of these original examples provide the motivation for much of what is being done, and has been done, in planar nearrings. Almost concurrently, Giovanni Ferrero was concerned with constructing finite integral nearrings. It turned out that his nearrings were planar. Ferrero then gave a new and very important direction to the study of planar nearrings. Parallel to the geometric interpretation of the algebraic structure of Anshel’s examples, Ferrero showed that some finite planar nearrings could be used to construct new balanced and partially balanced incomplete block designs. He suggested that these might be useful for coding theory, and his insight has proven to be prophetic.
Parallel and Sequential Methods for Ordinary Differential Equations
Authors: Burrage Kevin 
Published: November 2023

Abstract
This book presents an up-to-date exposition of the current `state of the art' of numerical methods for solving ordinary differential equations in a parallel computing environment. Although the main focus is on problems of initial value type, consideration will also be given to boundary value problems and partial differential equations. Furthermore, because linear algebra is an important component of the solution of differential equations, a complete chapter is devoted to the parallel solution of linear systems of equations. In addition to presenting an overview of parallel computing in general, two chapters are devoted to a summary of existing sequential differential equation methods. The parallel techniques discussed include parallelism across the method, parallelism across the step, parallelism across the system, and dynamic iteration. The book concludes with a chapter on the behaviour of a parallel code based on waveform relaxation. This comprehensive book is unique in its content and provides a balance between theoretical and practical issues by providing general frameworks in which to study parallel methods.
Patterns, related to rational numbers; periodic decimal fractions, repunits, and visible lattice points
Authors: Cofman Judita 
Published: November 2023

Abstract
Numbers expressible as fractions a/b, where a and bare integers and b &gt; 0, are called rational numbers. Each rational number can be expanded as a decimal fraction (e.g. 2/5 = 0.4, 5/3 = 1.666 ... ) with either a finite or infinite number of decimal places. In Sections 3.1 and 3.2 we shall study sequences of digits in decimal expansions of rational numbers; we shall also deduce a divisibility criterion for repunits - important numbers in computer science and coding theory.
Two famous inequalities and some related problems
Authors: Cofman Judita 
Published: November 2023

Abstract
We shall focus attention on two classical inequalities, with vanous applications in modern mathematics: The isoperimetric theorem has a long history. It was formulated more than 2000 years ago; however its first rigorous proof was carried out as late as the nineteenth century. All known proofs of the isoperimetric theorem are far from elementary and cannot be presented in this book. We shall limit ourselves to giving a brief outline of the ideas behind the proof suggested by Jacob Steiner (1796-1863) and corrected by Karl Weierstrass (1815—1897). This will be done in Section 6.3. This section contains a number of problems dealing with special cases of the isoperimetric theorem.
[eq]
‘Mysteries’ of the third dimension: on cubes, pyramids, and spheres
Authors: Cofman Judita 
Published: November 2023

Abstract
Is it possible to double the cube - that is, to construct with compasses and straightedge the edge of a cube the volume of which. is twice that of a given cube? Are any two pyramids with congruent bases and congruent corresponding heights equidecomposable? (In other words, is it possible to dissect one of them into finitely many polyhedra, which can be assembled to form the other pyramid?) Can thirteen congruent solid spheres touch simultaneously a given sphere of the same size? The above questions became famous because they occupied mathematicians for centuries; finally they could be answered each of them with a ‘no’.
From puzzles to research topics: selected problems of combinatorics
Authors: Cofman Judita 
Published: November 2023

Abstract
Various puzzles prompted important research in combinatorics and related fields of mathematics. Euler’s 36 officers problem is a famous example: 36 officers of 6 ranks and from 6 regiments have to be arranged in a 6 x 6 square formation so that every row and every column of the formation includes officers of each rank and each regiment. Euler’s study of the above puzzle initiated the study of Latin squares which nowadays play an important role in statistics and in various branches of discrete mathematics. Our aim in the first part of this chapter is to explain the connection between the puzzle and Latin squares, to solve problems on Latin rectangles and Latin squares (in Section 10.1), and to point out an important link between Latin squares and a class of non-Euclidean geometries the finite projective planes (in Section 10.2).
Transformations And Steep Fronts
Authors: Baines M J 
Published: November 2023

Abstract
In this chapter we look at the moving grid problem from the so-called Lagrangian point of view and re-write the MFE procedure in new variables. We then introduce the Legendre transformation which plays a central role in the structure of the MFE method, at least for first-order PDE’s. Finally, we discuss the analytic behaviour of shocks and steep fronts, whose numerical description is the motivation for MFE.
Exact Semi-Discrete Mfe Solutions
Authors: Baines M J 
Published: November 2023

Abstract
Using the Lagrangian point of view, we can single out a class of first-order PDEs for which the residual minimization part of MFE is unnecessary, simply because the residual can be made identically zero (Baines 1985c, Baines 1985a, Baines 1991). This is a consequence of the fact that, for this class of PDEs, the driving function ℒU already lies in the space spanned by the α’s and β’s, called the tangent plane of the approximating manifold (see (Miller and Miller 1981)). For these PDEs, therefore, no L2 projection is required and approximate solutions can be constructed directly without residual minimization.
Mfe And Moving Best Fits
Authors: Baines M J 
Published: November 2023

Abstract
As we have seen in Chapter 9, the advantage of studying best fits to continuous functions by piecewise discontinuous linear or constant functions with variable nodes is that the inherent nonlinearities in the calculation of the nodal positions are much easier to handle. Moreover the steps in the iteration procedure of the algorithm generating these fits are completely local and singularities can be avoided. We shall use this study to obtain a new perspective on MFE, with the possibilities of avoiding singularities altogether and devising an extension of MFE to piecewise constant functions.
Sums Of Squares And Gauss Sums
Authors: Rose H E 
Published: November 2023

Abstract
The concepts introduced so far form the basis for the major topics to be discussed in this chapter, they are Gauss sums and, to begin with, the representation of integers as sums of squares. Consider the proposition: if p is a prime and p = 1 (mod 4), then the Diophantine equation has an integer solution. This result. which is generally attributed to Fermat, is one of the seminal theorems of number theory. Both the ideas associated with its many proofs and the study of its extensions and generalizations have had a major influence on the development of the subject. We shall discuss the infinite descent method, and then use it to prove Fermat’s result and some related results including Lagrange’s four square theorem. In Chapters 9 and 10 we shall treat the general arithmetic theory of quadratic forms.
Quadratic Forms
Authors: Rose H E 
Published: November 2023

Abstract
Gauss was the first to study quadratic forms in a systematic way, his famous book published in 180 l and called Disquisiriones A.rirhmeticae (see Gauss 1966). is largely taken up with this topic. Although he brought together a number of results from earlier mathematicians. many of the basic ideas and theorems are due to him and were first presented in Disquisitiones Arithmeticae. For example. he realized the fundamental importance of the equivalence of forms (Section 1) and introduced the notation of composition which is central to the theory of binary forms (Chapter 10).
Quasistatic cycles in dynamic linear thermoelasticity
Authors: Day W A 
Published: November 2023

Abstract
This paper is prompted by considerations drawn from thermodynamics. The origins of thermodynamics lie in attempts to understand the operations of heat engines. Reduced to its essentials, a heat engine is a deformable body immersed in a heat bath whose temperature is periodic; the body is subjected to a periodically-varying pressure whose period coincides with that of the heat bath, and, in practice, the pressure is adjusted so as to extract a positive amount of work in each cycle that the body performs. Because this situation is central within thermodynamics it is of considerable interest to develop techniques which will determine exactly, or at least provide tolerable approximations to, the temperature and displacement fields within the body.
Linear dynamical stability in constrained thermoelasticity II. Deformation-entropy constraints
Authors: Scott N H 
Published: November 2023

Abstract
The theory of infinitesimal disturbances of a uniform reference configuration Be of a constrained heat-conducting elastic body, developed in part I, is adapted here to the situation in which the constraint links the deformation and the entropy. There are now only three modes of plane-harmonic-wave propagation and, in contrast to the findings of part I, they all turn out to be linearly stable under conditions of a conventional kind on the material constants in Be. An a priori case is thereby established for the acceptability in thermomechanics of this type of constraint. The properties of the modes are investigated in some detail and compared with the corresponding solutions in the absence of a constraint. A limiting procedure is formulated which yields, as extreme cases, the secular equations for the constrained and unconstrained bodies.
A Construction for Groups
Authors: Hoffman P N , Humphreys J F 
Published: November 2023

Abstract
In this chapter, we shall describe a binary operation within a certain class of objects, each of which is a group with extra structure. This construction can he used to determine the structure of the centralizer of an element of, and so we shall be able to deduce information about the conjugacy classes of S˜n. In later chapters, the construction will play a role in the definition of certain representations.
Software for shape-preserving spline interpolation
Authors: Reinsch Christian H 
Published: November 2023

Abstract
Interpolation and smoothing of discrete data belong to the basic tasks of numerical analysis like solving linear equations and eigenvalue problems, or finding polynomial roots. The successful technique of using piecewise polynomials as a basis is theoretically justified by a minimum principle. This remains valid if shape preservation is added as a constraint, although the construction process is then non-linear. We discuss robust algorithms with good convergence rates, to be used in public program libraries.
Subharmonic and plurisubharmonic functions
Authors: Klimek Maciej 
Published: November 2023

Abstract
In this chapter we begin our study of plurisubharmonic functions; they will play the central part throughout the remaining chapters. First, however, we present a comprehensive account of some fundamental properties of harmonic, pluriharmonic, and subharmonic functions. We also give some examples of applications of these functions in complex analysis. This is followed by a short exposition of some fundamental properties of plurisubharmonic functions. The analogies between plurisubharmonicity, pseudoconvexity, and convexity (of functions and sets, respectively) are explored in the exercises closing the chapter.
Maximal functions with logarithmic singularities
Authors: Klimek Maciej 
Published: November 2023

Abstract
The aim of this chapter is to show that in Cn one can introduce pluricomplex Green functions which generalize the one-variable Green functions with logarithmic poles. In the course of our study of pluricomplex Green functions it will be necessary to apply the Monge-Ampere operator to some unbounded plurisubharmonic functions. In this particular case, only a very simple modification of the definition of the Monge-Ampere operator will be required.
The Second World War: the emergence of government computing centres
Authors: Croarken Mary 
Published: November 2023

Abstract
The work of Comrie, Hartree, and Lennard-Jones during the 1930s had done much to develop both calculating machines and computing techniques. Hartree and Lennard-Jones had also been instrumental in providing computing resources for particular groups of scientists. However, by 1939 the only organization to which scientific workers in government research establishments could apply for computational advice or practical assistance was Comrie’s Scientific Computing Service (the SCS).
Post-war computing service centres
Authors: Croarken Mary 
Published: November 2023

Abstract
The NPL Mathematics Division was set up to act as a national computing centre to serve most of Britain’s computing needs. It epitomized the centralization concept both by providing computing services of all kinds and by carrying out research into new computing machinery and numerical techniques. The existence of the NPL Mathematics Division did not prevent several other institutions from developing computing centres of their own. Perhaps the most well-known British scientific computing service centre, other than the NPL Mathematics Division, was the Scientific Computing Service run by L. J. Comrie.
Monotone logic programming*
Authors: Blairt† Howard A , Brown Allen L , Subrahmanian§ ‡ Jr. V S 
Published: November 2023

Abstract
We propose a notion of an abstract logic. Based on this notion, we define abstract logic programs to be sets of sentences of an abstract logic. When these abstract logics possess certain logical properties (some properties considered are compactness, finitariness, and monotone consequence relations) we show how to develop a fixed point, model-state-theoretic and proof theoretic semantics for such programs. The work of Fitting on developing a generalized semantics for multivalued logic programming is extended here to arbitrary abstract logics. We present examples to show how our semantics is robust enough to be applicable to various non-classical logics like temporal logic and multivalued logics, as well as to extensions of classical logic programming such as disjunctive logic programming. We also show how some aspects of the declarative semantics of distributed logic programming, particularly the work of Ramanujam, can be incorporated into our framework.
Basic defeasible logic
Authors: Nute* Donald 
Published: November 2023

Abstract
Defeasible logics are nonmonotonic formalisms. A consistent defeasible theory can contain the defeasible rule A⇒ p, the antecedent A of the rule, and the complement ,p of its consequent. Conflicts between defeasible rules with incompatible consequents are resolved using an explicit superiority relation on rules or a method for deter mining which of two rules has the more specific antecedent. Proofs are trees whose nodes are labelled by defeasible theories, wffs of the logic, and a + or a – to show that the wff is derivable or is demonstrably not derivable from the theory. Defeasible logics are defined as sets of conditions on nodes of proof trees. A family of defeasible logics is described in which familiar examples of nonmonotonic reasoning can be represented and intuitively correct results can be derived from them. These defeasible logics are decidable.
Problem solving and theorem proving
Authors: Hogger Christopher John 
Published: November 2023

Abstract
The problem-solving mechanism of logic programming is fundamentally a theorem proving (inferential) mechanism. As indicated earlier in the course, our aim is to characterize computation as the inference-driven manipulation of knowledge. More concretely, the interpreters that we employ to execute logic programs are just automated theorem provers, so that we might expect proof theory to tell us something about their capabilities and limitations.
The Herbrand domain
Authors: Hogger Christopher John 
Published: November 2023

Abstract
When we compose a logic program we do so with some 'real—world' domain in mind. But, as remarked earlier, the computer cannot (normally) operate directly upon that domain—instead, its workings are confined to its own internal domain comprising just the symbols representable within it. In the logic programming context that domain is called the Herbrand domain (or Herbrand universe), and is determined wholly by the language (set of sentences) L in which we formulate our programs and queries. Accordingly, we denote it either by HL or—when the choice of language is clear in the current context-by—just H.
Inference systems for clausal form
Authors: Hogger Christopher John 
Published: November 2023

Abstract
In this Theme we take a preliminary look at the issue of how one might set up an automated problem solver capable of investigating problems formulated in clausal-form logic.
It is worth observing straightaway that clausal-form logic can be approximated by the much simpler system of propositional (variable-free) logic. The approximation is facilitated by recognizing that only the Herbrand domain H need be considered. This is because any model for a clause-set over some realworld domain necessarily has an isomorphic (structurally-identical) model over H-in effect, those two models are just renamings of one another. According to the approximation, any clause of the form is shorthand for a set of implicitly-conjoined ground clauses
Resolution for propositional logic
Authors: Hogger Christopher John 
Published: November 2023

Abstract
Resolution comprises a single inference rule applicable to clausal-form logic. Briefly, from any two clauses having an appropriate form, resolution derives a new clause as their consequence. It is best explained by first showing how it works in the special case of propositional clausal form.
Therefore suppose that we have any set of propositional clauses. A resolution step entails first selecting any two of these clauses, called the parents for this step, constrained only by the requirement that some literal A in one parent be complementary to some literal -,A in the other parent (two literals are complementary when one is the negation of the other). Thus the parents must take the forms:
Extending resolution to predicate logic
Authors: Hogger Christopher John 
Published: November 2023

Abstract
In order to deal with clauses of predicate logic, the propositional resolution inference rule described in Theme 26 must be supplemented by the implicit use of a further sound inference rule known as universal instantiation:
Here is a simple example, in which the clauses’ assumed outermost universal quantifiers are, as usual, omitted from the presentation. Given the parent clauses
The ‘likes’ literals that we wish to make complementary by applying the substitution then refer respectively to two subsets of HxH. The substitution 0 effectively identifies the intersection of these two subsets, and it is this that is then referred to by the complementary ‘likes’ literals, as indicated in Figure 28.1.