bluuebunny/crossref_metadata_2025_split huggingface dataset:
Computation Of Gauss-Kronrod Quadrature Rules (With D. Calvetti, W.B. Gragg And L. Reichel)
Authors: Chan Raymond H , Greif Chen , O’Leary Dianne P 
Published: November 2023

Abstract
If I think about the genesis of this paper, my semester at Stanford in 1995 comes to mind right away. For the mathematical orphan that I was, since my advisor, Peter Henrici died before I completed my PhD, the opportunity to spend a semester at Stanford was a truly unique opportunity. As Gene and I team-taught a seminar course on quadrature rules, linear algebra, and large-scale computations, I discovered more charming aspects of quadrature rules than I thought there could be. I must admit that when Gene first talked about the importance of Gauss–Kronrod rules in estimating errors in numerical integration, I thought that, for sure, I would never be caught working on such a dull topic. Well, soon enough I started seeing quadrature rules everywhere, and their elegance and secret connections with nearly everything began to get hold of me. Even Gauss–Kronrod quadrature rules, until then quickly dismissed even from my teaching, started to show their discrete charm. Well, never say never!
Ill-Conditioned Eigensystems And The Computation Of The Jordan Canonical Form (With J. H. Wilkinson)
Authors: Chan Raymond H , Greif Chen , O’Leary Dianne P 
Published: November 2023

Abstract
The solution of the complete eigenvalue problem for a no normal matrix A presents severe practical difficulties when A is defective or close to a defective matrix. Moreover, in the presence of rounding errors, one cannot even determine whether or not a matrix is defective. Several of the more stable methods for computing the Jordan canonical form are discussed, together with the alternative approach of computing well-defined bases (usually orthogonal) of the relevant invariant subspaces.
Liapunov methods for determining stability of the zero solution
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
In Chapter 1, we described the ‘energy method ‘ for determining that an equilibrium point at the origin is an attractor, in connection with simple systems that allow a discussion in terms of the change in energy levels along the phase paths as time progresses. The Liapunov theory described in this chapter may be regarded as a development of this idea. It has no necessary connection with energy, concerns solution stability rather than Poincaré stability, and has a broader field of application. We discuss only two-dimensional autonomous systems in detail, but the theory can be extended to higher-dimensional and nonautonomous cases (Cesari 1971).
Second-order differentialequations in the phase plane
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
1.1. For the general equation x f (x, x), (see eqn (1.6)), equilibrium points lie on the x axis, and are given by all solutions of f (x, 0) 0, and the phase paths in the plane (x, y) (y x) are given by all solutions of the first-order equation Note that scales on the x and y axes are not always the same. Even though explicit equations for the phase paths can be found for problems (i) to (viii) below, it is often easier to compute and plot phase paths numerically from x f (x, x), if a suitable computer program is available. This is usually achieved by solving x y, y f (x, y) treated as simultaneous differential equations, so that (x(t), y(t)) are obtained parametrically in terms of t . The phase diagrams shown here have been computed using Mathematica.
Stability
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
Poincaré or orbital stability is defined by Definition 8.1. A general descriptive or graphical approach is adopted in Problems 1–4, which are concerned with Poincaré stability. Consider the stability of a typical closed path , within any strip bounded by two nearby closed paths (shown shaded). All half-paths starting in the strip remain in it for all time, so is Pincaré stable. The same applies to any typical path in the region describing (periodic) whirling motion beyond the separatrices. The separatrices are not Poincaré stable, since there are neighbouring half-paths that deviate unboundedly from any separatrix.
Exploring Complex Data Sets
Authors: Krzanowski Wojtek J 
Published: November 2023

Abstract
Complex data sets can arise in a variety of ways. One is when the sample size is relatively small but many interrelated variables are measured on each individual, so that the associations between the variables obscure any patterns that might exist among the individuals. In other situations, fewer variables may be present but sample sizes may be too big for easy assimilation. In yet other cases, of course, we may have very many variables as well as large samples. The collection of large data sets is now commonplace, partly because of the development of increasingly sophisticated recording devices and partly because it is possible to store massive data sets electronically in such a way that instant retrieval and sorting is no problem.
The archaeology of mathematics in an ancient Greek city
Authors: Romano David Gilman 
Published: November 2023

Abstract
A modern mapping and research project, the Corinth Computer Project, has been underway in ancient Corinth in Greece, as well as in Philadelphia, for the past twenty years, 1988–2007. By means of highly accurate electronic total station surveys in the field, and advanced digital cartographic techniques in the laboratory, one of the project objectives has been to analyze the ways in which Greek and Roman surveyors worked, both in the city of Corinth and in the surrounding landscape.
Patronage of the mathematical sciences in Islamic societies
Authors: Brentjes Sonja 
Published: November 2023

Abstract
Patronage is a term that has no direct equivalent in any of the languages in which mathematical sciences were practised in Islamic societies. If we take one of the ancient Roman meanings of patronage, namely the existence of a clientele relationship between a protector and a protégé, as the starting point for exploring social relationships involving the mathematical sciences in Islamic societies, we will see that practitioners of these sciences could indeed enter relationships with people of higher social status and greater wealth, which provided them with relatively stable positions and various benefits.
Human computers in eighteenthand nineteenth-century Britain
Authors: Croarken Mary 
Published: November 2023

Abstract
In eighteenthand nineteenth-century Britain, people who earned their living either by calculating mathematical tables or by undertaking other computational work were referred to as computers or calculators (see Simpson and Weiner 1989). Today we associate these terms with electronic or mechanical devices, but there are strong parallels between the modern use of the word computer and its application to people two hundred years ago. The most common sphere in which human computers worked was astronomy, but they were also employed in a variety of scientific fields on a lesser scale.
Mixing, building, and feeding: mathematics and technology in ancient Egypt
Authors: Rossi Corinna 
Published: November 2023

Abstract
The number of surviving mathematical sources from ancient Egypt is relatively small, but they all give a consistent picture of mathematics as deeply intertwined with a variety of practical activities. Mathematics, after all, is a constant presence in the daily life of any population, even if in different forms or contexts (Selin 2000; Cuomo 2001; Asper, Chapter 2.1 in this volume). Beside the presence of geometrical patterns in decorative or functional objects (e.g., Robson 2000; Wendrich 2000; Whitley 2001, 77–133; Brezine, Chapter 5.4 in this volume), counting and measuring are naturally embedded in a large number of technological activities.
Sanskrit mathematical verse
Authors: Plofker Kim 
Published: November 2023

Abstract
The word ‘mathematics’ nowadays inspires an expectation of numbers and formulas visually presented on a written page. The dominance of literacy in modern learning makes it di7cult for us to conceive of advanced mathematical knowledge in any non-literate (or even non-symbolic) format. In ancient and medieval Indian cultures, on the other hand, the veneration of spoken Sanskrit produced an ideal of oral learning that embraced technical subjects including the mathematical sciences, as well as narrative literature and belles lettres.
Mathematics, music, and experiment in late seventeenth-century England
Authors: Wardhaugh Benjamin 
Published: November 2023

Abstract
The Scientific Revolution saw many subjects given new scrutiny, with attempts to use mathematical, mechanical, or experimental modes of explanation to gain understanding of them. One of those subjects was music. Already a tradition of mathematical study of musical intervals stretched back through the middle ages to ancient Greece, where the emphasis had been on ratios of the lengths of strings that formed particular musical intervals. In the seventeenth century there were new mathematical techniques and new kinds of mechanical explanation that could be applied instead (Wardhaugh 2006).
Modernism in mathematics
Authors: Gray Jeremy 
Published: November 2023

Abstract
There were dramatic changes in several branches of mathematics between 1880 and 1920. Among the best documented are the rise of modern algebra, associated with the work of Emmy Noether and her followers and which she traced back to the work of Richard Dedekind; Henri Lebesgue’s axiomatic theory of the integral; the introduction of algebraic topology by Henri Poincaré; and the new axiomatic geometry first developed by the Italian school around Guiseppe Peano and later by David Hilbert.
‘Gigantic implements of war’: images of Newton as a mathematician
Authors: Guicciardini Niccolò 
Published: November 2023

Abstract
That Newton was a great mathematician was evident in the early 1670s to those very few who had privileged access to his early mathematical discoveries. By the 1680s Newton was regarded by his contemporaries as an outstanding mathematician. It was in order to find an answer to a mathematical question concerning the shape of planetary orbits, for example, that Edmond Halley travelled to Cambridge to see Newton in August 1684. Th ere he found that Newton had already broached and answered that di7cult question.
From cascades to calculus: Rolle’s theorem
Authors: Barrow-Green June 
Published: November 2023

Abstract
Rolle’s theorem is a simple but important result, familiar to anyone who has moved just beyond elementary calculus into the beginnings of analysis. Essentially it tells us that if a differentiable function has equal values at a and b , then somewhere between those two points it must have a local maximum or a local minimum (Fig. 8.3.1). A more formal statement of the theorem, typical of those given in modern textbooks, is as follows.
Reading ancient Greek mathematics
Authors: Saito Ken 
Published: November 2023

Abstract
Since the history of mathematics tells narratives about past mathematics, it is obvious that all that it narrates is based on one or more source documents, for nobody among us has ever met Archimedes, or even Gauss. However, the source documents are not in themselves history, and the fundamental task of the historian is to make a history from those documents. There are particular features and di7culties in the interpretation of sources for the history of mathematics, which depend on the period and area treated.
Cluster Allocation Design Networks*
Authors: Madrigal Ana Maria 
Published: November 2023

Abstract
Policy makers usually define a strategy that involves policy assignment and recording mechanisms. Causal inference is sensitive to the specification of these mechanisms. Influence diagrams have been used for causal reasoning within a Bayesian decision-theoretic framework (Dawid, 2002). Design Networks (DNs) expand this framework by including experimental design decision nodes. DNs provide semantics to discuss how a design decision strategy might assist the identification of intervention causal effects. The DN framework is extended to Cluster Allocation. Cases of ‘pure’ cluster (all individuals in a cluster receiving the same intervention) and ‘non-pure’ cluster (only a subset receiving the policy) are discussed in terms of causal effects.
Ergodic Averages Via Dominating Processes*
Authors: Møller Jesper , Mengersen Kerrie 
Published: November 2023

Abstract
We show how the mean of a monotone function (defined on a state space equipped with a partial ordering) can be estimated, using ergodic averages calculated from upper and lower dominating processes of a stationary irreducible Markov chain. In particular, we do not need to simulate the stationary Markov chain and we eliminate the problem of whether an appropriate burn-in is determined or not. Moreover, when a central limit theorem applies, we show how confidence intervals for the mean can be estimated by bounding the asymptotic variance of the ergodic average based on the equilibrium chain.
Stochastic Integration With Locally Square-Integrable Martingales
Authors: Medvegyev Péter 
Published: November 2023

Abstract
In this chapter we shall present a relatively simple introduction to stochastic integration theory. Our main simplifying assumption is that we assume that the integrators are locally square-integrable martingales. Every continuous process is locally bounded, hence the space H contains the continuous local martingales. In most of the applications the integrator is continuous, therefore in this chapter we shall mainly concentrate on the continuous case. As we shall see, the slightly more general case, when the integrator is in His nearly the same as the continuous one.
The Structure Of Local Martingales
Authors: Medvegyev Péter 
Published: November 2023

Abstract
The main result of this chapter is the theorem which we shall call the Fundamental Theorem of Local Martingales, which states that every local martingale can be decomposed as the sum of a locally bounded local martingale and a local martingale which has locally integrable variation. The Fundamental Theorem of Local Martingales has many important consequences. Perhaps the most important one is that for any local martingale L the quadratic variation process [L] exists and the difference L− [L] is a local martingale.
General Theory Of Stochastic Integration
Authors: Medvegyev Péter 
Published: November 2023

Abstract
In this chapter we discuss the general theory of stochastic integration. We shall assume that the integrators are semimartingales, but we shall not assume that the integrands are locally bounded. In the first part of the chapter we shall prove that every local martingale is the sum of a continuous and a purely discontinuous local martingale. One can think about purely discontinuous local martingales as sums of continuously compensated single jumps. The quadratic variation of a purely discontinuous local martingale is the sum of the squares of the jumps of the local martingale.
General relativity
Authors: Rendall Alan D 
Published: November 2023

Abstract
In this section some of the basic concepts of general relativity are introduced. First some necessary linear algebra is presented. Let V be a finite dimensional real vector space. A vector in V can be represented by its components vα in a given basis. The dual space Vα consists of the linear functionals on V, i.e. the linear mappings from V to the real numbers ℝ.
Matter models
Authors: Rendall Alan D 
Published: November 2023

Abstract
General relativity models the interaction of material bodies due to the gravitational field. In addition to the Einstein equations which describe the field a description of the matter is necessary. There are many different matter models of relevance to different physical situations and in this chapter a number of the most important of these are surveyed. To specify a matter model the following three components must be fixed: the nature of the mathematical objects (matter fields) which describe the particular type of matter; the form of the energy-momentum tensor as a function of the basic matter fields and the metric; the equations of motion of the matter - these are partial differential equations for the matter fields which also depend on the metric.
Symmetry classes
Authors: Rendall Alan D 
Published: November 2023

Abstract
The set of all isometries of a pseudoriemannian metric forms a group under composition, the isometry group. In fact it is a Lie group. This means that it admits the structure of a manifold in such a way that the group operations (composition and taking the inverse) are smooth. The generators of one-parameter subgroups of a Lie group G form a vector space of the same dimension as the group. Taking the Lie brackets of two of these vector fields gives another vector field of the same type. In this way the vector space of generators of one-parameter subgroups gets an algebraic structure which makes it into a Lie algebra.
Elliptic equations
Authors: Rendall Alan D 
Published: November 2023

Abstract
This chapter is an introduction to elliptic equations. These often provide models for equilibrium situations in physics or other sciences. In general relativity they are also important as a tool for studying the set of solutions of the constraints. Another application is to the equations defining certain coordinate conditions. In this section some basic concepts are introduced which are useful in the classification of systems of partial differential equations.
Biosecurity and quarantine for preventing invasive species
Authors: Maynard Glynn , Nowell David 
Published: November 2023

Abstract
The saying ‘prevention is better than cure’ applies to the entry of invasive species but this is difficult to achieve, particularly in the absence of physical or ecological barriers to the movement of invasive species, or where human activities and vectors provide pathways for their entry. For the purposes of this chapter, the term ‘invasive species’ applies to those that enter and establish in a new area and have the ability to spread aggressively, to intrude or overwhelm other organisms.
The Evolution of EU Citizenship
Authors: Maas Willem 
Published: November 2023

Abstract
Although the formal category of EU citizenship was introduced with the Maastricht Treaty (1993), the extent to which the term reflects reality is unclear. In its most common definition, citizenship is composed of rights, which usually evolve through a process of contestation among individuals, groups, and institutions, shaped by and in turn transforming the political culture. European citizenship emerged out of a series of critical political junctures that span the entire history of European integration, coupled with an ongoing institutional and political commitment to safeguarding and promoting the development of European rights. Initially, the freer trade of the 1950s and 1960s created demands for freer movement of labor.
Congruences To Composite Moduli
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
Linear congruences. We have supposed since § 7.4 (apart from a momentary digression in § 7.8) that the modulus mis prime. In this chapter we prove a few theorems concerning congruences to general moduli. The theory is much less simple when the modulus is composite, and we shall not attempt any systematic discussion. We considered the general linear congruence in § 5.4, and it will be convenient to recall our results. The congruence is insoluble unless If this condition is satisfied, then (8.1.1) has just dsolutions, viz. of linear congruences to coprime moduli m1, m2, ..., mk. The system will be insoluble unless (ai, mi) bifor every i.
The Fundamental Theorem Of Arithmetic
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
Algebraic numbers and integers. In this chapter we consider some simple generalizations of the notion of an integer. We defined an algebraic number in § 11.5; ξis an algebraic number if it is a root of an equation then ξ is said to be an algebraic integer. This is the natural definition, since a rational ξ a/b satisfies bξ a 0, and is an integer when b 1. When n2, ξis said to be a quadraticnumber, or integer, as the case may be. These definitions enable us to restate Theorem 45 in the form Theorem 206. An algebraic integer, if rational, is a rational integer. † We defined the ‘rational integers’ in § 1.1. Since then we have described them simply as the ‘integers’, but now it becomes important to distinguish them explicitly from integers of other kinds.
The Order Of Magnitude Of Arithmetical Functions
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
The order of d(n). In the last chapter we discussed formal relations satisfied by certain arithmetical functions, such as d(n), ς (n), and φ(n). We now consider the behaviour of these functions for large values of n, beginning with d(n). It is obvious that d(n) 2 when n &gt;1, while d(n)= 2 if nis a prime. Hence It is less trivial to find any upper bound for the order of magnitude of d(n).
Partitions
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
The general problem of additive arithmetic. In this and the next two chapters we shall be occupied with the additive theory of numbers. The general problem of the theory may be stated as follows. Suppose that Ao is a given system of integers. Thus Amight contain all the positive integers, or the squares, or the primes. We consider all possible representations of an arbitrary positive integer nin the form where smay be fixed or unrestricted, the amay or may not be necessarily different, and order may or may not be relevant, according to the particular problem considered. We denote by r(n)the number of such representations. Then what can we say about r(n)? For example, is r(n)always positive? Is there always at any rate one representation of every n? Partitions of numbers. We take first the case in which Ais the set 1, 2, 3, ...of all positive integers, sis unrestricted, repetitions are allowed, and order is irrelevant. This is the problem of ‘unrestricted partitions’. A partitionof a number nis a representation of nas the sum of any number of positive integral parts. Thus has 7 partitions.† The order of the parts is irrelevant, so that we may, when we please, suppose the parts to be arranged in descending order of magnitude. We denote by p(n)the number of partitions of n; thus p(5)= 7. We can represent a partition graphically by an array of dots or ‘nodes’ such as † We have, of course, to count the representation by one part only.
The Representation Of A Number By Two Or Four Squares
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
Waring’s problem: the numbers g(k)and G(k). Waring’s problem is that of the representation of positive integers as sums of a fixed number sof non-negative kth powers. It is the particular case of the general problem of § 19.1 in which the aare and sis fixed. When k1, the problem is that of partitions into sparts of unrestricted form; such partitions are enumerated, as we saw in Ch. XIX, by the function it is plainly impossible to represent all integers if sis too small, for example if s= 1. Indeed it is impossible if s &lt; k.
Democracy and Growth: A Case Study in Failed Causal Inference
Authors: Seawright Jason 
Published: November 2023

Abstract
Political scientists have seen the relationship between democracy and socioeconomic development as a key analytic puzzle at least since the publication of Seymour Lipset’s classic study (1959) of the social prerequisites of democracy. In recent decades, many analysts in this literature have studied the question of whether democracy has a causal effect on economic growth. As discussed below, the literature on this research question is both lively and voluminous. Yet this literature raises important concerns because researchers have been unable to even approximate a consensus on the central question of whether democracy causes economic growth.