bluuebunny/crossref_metadata_2025 huggingface dataset:
spatio-Temporal Properties of Gene Genealogies in Geographically Structured Populations
Authors: Epperson Bryan K
Published: November 2023

Probabilities of identity by descent (IBD) in geographically structured populations incorporate genealogical information for genetic markers in processes of genetic drift and migration. Existing theory has dealt only with the spatial or geographic pattern of degrees of relatedness or genetic similarity. However, to fully understand the relationship of present geographical or spatial patterns of genetic variation to those in the past, it is necessary to build models that directly relate the two. The formulation of space-time probabilities of IBD is very general and can be applied to a wide range of systems of populations exchanging migrants, and it can be applied to experimental studies to infer migration patterns that occurred in the past. In this chapter, results are presented that illustrate several features of the historical relationships between genes in different populations. Coalescence probabilities for two genes, separated in time as well as space, are also examined. The results can be used to determine the level of certainty that genes found in present populations are descended from ancient genes in the same population or nearby populations versus geographically distant populations. When mutation and migration rates are high, past populations that are quite distant geographically are almost as likely to contain ancestors of (and hence show almost equivalent similarity with) genes at a given population as is the past population at the same location, and thus it should be difficult to infer the geographic origins of genetic polymorphisms. Space-time probabilities of IBD also can be used for’space time’ molecular genetic data, which is becoming increasingly important as DNA polymorphism is increasingly being assayed from fossils and other sources of ancient DNA.
The Geometric Picture
Authors: Higgins Peter M
Published: November 2023

The two sides of the mathematical coin, discrete mathematics based on counting, as opposed to the continuous mathematics that arises through measurement, still form the basis of modern mathematics and the tension between the two branches persists today. The nature of this tension first emerged clearly during the lifetime of Pythagoras in the 6th century BC for it was at this time, in ancient Greece, that arithmetic and geometry reached the level of sophistication required to formulate the difficulties.
Old English Literature
Authors: Sanders Andrew
Published: November 2023

The term ‘Old English’ was invented as a patriotic and philological convenience. The more familiar term ‘Anglo-Saxon’ has a far older pedigree. ‘Old English’ implied that there was a cultural continuity between the England of the sixth century and the England of the nineteenth century (when German, and later British, philologists determined that there had been phases in the development of the English language which they described as ‘Old’, ‘Middle’, and ‘Modern’). ‘Anglo-Saxon’ had, on the other hand, come to suggest a culture distinct from that of modern England, one which might be pejoratively linked to the overtones of’ Sassenach’ (Saxon), a word long thrown back by angry Celts at English invaders and English cultural imperialists. In 1871 Henry Sweet, the pioneer Oxford phonetician and Anglicist, insisted in his edition of one of King Alfred’s translations that he was going to use ‘Old English’ to denote ‘the unmixed, inflectional state of the English language, commonly known by the barbarous and unmeaning title of “Anglo-Saxon”‘. A thousand years earlier, King Alfred himself had referred to the tongue which he spoke and in which he wrote as ‘englisc’. It was the language of the people he ruled, the inhabitants of Wessex who formed part of a larger English nation. That nation, which occupied most of the fertile arable land in the southern part of the island of Britain, was united by its Christian religion, by its traditions, and by a form of speech which, despite wide regional varieties of dialect, was already distinct from the ‘Saxon’ of the continental Germans. From the thirteenth century onwards, however, Alfred’s ‘English’ gradually became incomprehensible to the vast majority of the English-speaking descendants of those same Anglo Saxons. Scholars and divines of the Renaissance period may have revived interest in the study of Old English texts in the hope of proving that England had traditions in Church and State which distinguished it from the rest of Europe.
Medieval Literature 1066—1510
Authors: Sanders Andrew
Published: November 2023

Strictly speaking, the Bayeux Tapestry, which provides the most vivid pictorial record of the events leading up to the conquest of England by the Normans, is not a tapestry at all. The 70-metre long embroidery, known in the Norman cathedral city of Bayeux as ‘the tapestry of Queen Matilda’, is equally unlikely to be the painstaking work of the wife of William the Conqueror. Long before the Conquest, and long after it, England was famed for the intricacy and brilliance of its needlework. The great narrative hanging was probably the result of a celebratory, and possibly enforced, commission to English needle women to mark both the Norman victory of rn66 and the consecration of the cathedral at Bayeux in 1077 by its bishop, William’s half-brother Odo. After the conquest Odo had been rewarded by William with large estates in England and with the title Earl of Kent. He later acted, with some ruthlessness, as the King’s viceroy in the north of England. Odo’s periodic and prominent appearances on the tapestry as William’s counsellor, as the blesser of food at a banquet on English soil before the battle of Hastings, and as the armed wielder of a great wooden staff in the battle itself (clerics were forbidden to carry swords), suggest that he at least would not have found it inappropriate to decorate his new cathedral with an embroidered commemoration of his brother’s famous victory.
Modernism and its Alternatives: Literature 1920—1945
Authors: Sanders Andrew
Published: November 2023

When the architect Sir Edwin Lutyens visited the battlefields of north-eastern France in July 1917 in order to investigate the need for permanent memorials to the vast numbers of British and Empire war-dead, he described in a tone of horrified amazement the strange interaction of man and nature he had witnessed. ‘What humanity can endure and suffer is beyond belief’, he told his wife, ‘the battlefields-the obliteration of all human endeavor and achievement and the human achievement of destruction is bettered by the poppies and wild flowers that are as friendly to an unexploded shell as they are to the leg of a garden seat in Surrey.’ For the moment, the only monument he could envisage as appropriate was ‘a solid ball of bronze’. Lutyens’s ambiguity concerning what he had seen as a non-combatant on the Western Front proved typical of the future reactions of those remembering or contemplating the wasteful devastation of the war. The post-war period was haunted by long memories, some tender, some angry, most sickening. According to the critic and poet Herbert Read, writing in The Criterion in 1930, it had taken more than a decade for ex-combatants to come to terms with what the war had meant to them and with ‘the debris of its emotional conflicts’ before they could begin to transform their experience into literature. Nevertheless, it was the incongruity implicit in the idea of the ‘friendliness’ of the unfeeling wild flowers noted by Lutyens, as much as that of the equally indifferent poppies, cornflowers, skylarks and rats of the poetry that had emerged from the war, that effectively marked the end of an art which had once reached for comforting or sympathetic images from nature. Stark and solid balls of bronze seemed a more appropriate tribute to those sacrificed to the unfeeling might of the machines devised and exploited by human ingenuity.
Post—War and Post—Modern Literature
Authors: Sanders Andrew
Published: November 2023

When the Second World War ended in Europe in the summer of 1945, much of Britain was in ruins. Quite literally in ruins. Its devastated industrial cities were not exactly the heaps of rubble that appalled post-war visitors to Germany (Stephen Spender, for one, spoke of the ‘astonishing and total change, that incalculable shift from a soaring to a sinking motion which distinguishes a dead body’ that disturbed him when revisiting Hamburg), but British cities as diverse in character as Glasgow, Coventry, Canterbury, Bristol, Exeter, and Portsmouth had been torn apart by bombs. London, in particular, had been universally pitted and scarred and was now marked by absences where familiar landmarks had once stood. Whole districts were in ruins and most streets somehow bore the signs of blast, shrapnel, fire-bombs, or high explosives. Although its greater monuments, such as St Paul’s, had survived largely intact, the cathedral itself now rose hauntingly and, to some imaginative observers, resolutely above the shells of churches and blasted office buildings. This broken London of bricks, façades, and dangerously exposed basements can now be only recognized from paintings and photographs and from the cinematic exploitation of bomb-sites in films such as the two early comedies made at the famous Ealing Studios, Charles Crichton’s rowdy Hue and Cry (1947) and Harry Cornelius’s farce Passport to Pimlico (1949).
The Classical World
Authors: Arnaud P
Published: November 2023

The title ‘classical world’ usually indicates a situation that is limited in space and time and is characterized by the Greeks and by those who adopted their way of life after the fifth century BC. That is to say it applies in a strict sense to a very defined area, namely the shores of the Mediterranean. According to the terminology used by ancient writers, the classical world stood in contrast with ‘barbarian Europe’; however, contacts between these two expressions of Europe represented one of the dynamic features of the continent in antiquity.
Industrialization, 1740 to the Present
Authors: Pollard S
Published: November 2023

Industrialization should be understood as a technical term, carrying a particular meaning; it should not be taken to imply that before its onset there was no industry in Europe. The preceding chapter will have shown how much industry in fact there was. By general consent, the term has come to be applied to particular changes in industrial structure and technology, together with changes in other aspects of social life associated with them.
Franz Schubert (1797–1828)
Authors: Phillips Lois
Published: November 2023

My heart is heavy, my peace is gone; never, never shall I find it again.

Where he is not with me, to me is a grave; the whole world seems bitter as gall.

My poor head is in a frenzy, my poor mind shattered.

I seek only him as I peer from my window; to seek only him do I leave the house.

His proud bearing, his noble figure, his smiling lips, and compelling eyes,

the magic of his speech, the touch of his hand, and ah, his kiss!
Family
Authors: Caplow Theodore , Del Campo Salustiano
Published: November 2023

Three master trends—modernization, the gradual progress of equality, individuation—exerted a strong influence on both EU and US families between 1960 and 2000. The direct eﬀects of modernization include the steady decline of the infant mortality rate (deaths of infants in the first year of life), dramatic declines in fertility, corresponding decreases in average household size, and a vast improvement in the material equipment of family households. The gradual progress of equality greatly reduced the long-standing diﬀerences between men and women in sexual opportunity, access to the labor market, occupational careers, control of property, freedom to divorce, and relationships with children.
Curvature
Authors: Félix Yves , Oprea John , Tanré Daniel
Published: November 2023

Recently algebraic models have found a place in differential geometry, especially in questions revolving around types of curvature restrictions. In particular, F. Fang and X. Rong [84] and B. Totaro [256] have approached certain questions of Karsten Grove in this spirit and the work of Belegradek and Kapovitch [23] on the existence of non-negative curvature metrics on vector bundles has also relied on models. Of course, when we speak of algebraic models here, we are speaking of models ranging from simple cohomology all the way to minimal models and relative models.
Gene H. Golub Biography
Authors: Chan Raymond H , Greif Chen , O’Leary Dianne P
Published: November 2023

Gene Howard Golub was born on February 29th, 1932 to Bernice and Nathan Golub. His mother was from Latvia and his father from Ukraine. They both came to the United States independently of one another in 1923, and both settled in Chicago for family reasons: they each had an older sibling in the city.
Major Awards
Authors: Chan Raymond H , Greif Chen , O’Leary Dianne P
Published: November 2023

March, 1958, National Science Foundation Postdoctoral Fellowship for study at Cambridge University, Cambridge, England. [A2] Fellow of AAAS, 1981. [A3] Award for Leadership in Numerical Analysis (Forsythe Lecturer), 1978. [A4] Honorary Fellow, St. Catherine’s College, Oxford, 1983. [A5] Alumni Honor Award for Distinguished Service, University of Illinois at Urbana-Champaign, 1983.
An analysis of the total least squares problem (withC. Van Loan)
Authors: Chan Raymond H , Greif Chen , O’Leary Dianne P
Published: November 2023

This paper is included in this volume and was a thrill of read back in 1973 when I had just completed graduate school. Privately, I have always called the paper ‘Look How Far You Can Run with QR, SVD, and Schur’, a title that reminds me of Gene’s uncanny ability to turn difficult problems into tractable ones with elegant solutions. Gene’s SIREV paper is filled with ideas that kept the whole field busy for years: generalized eigenproblems, inverse eigenproblems, rank-1 modified eigenproblems, and of course, total least squares (TLS). TLS is there as an example of an eigenproblem that is buried inside a least squares problem and the paper we wrote was one of several that we did during the production of the first edition of our book.
The Simplex Method Of Linear Programming Using Lu Decomposition (With R. H. Bartels)
Authors: Chan Raymond H , Greif Chen , O’Leary Dianne P
Published: November 2023

This paper is a model for the way that Gene likes to work: find someone interesting in another field to talk with, find out what the problems, approaches, and computational practices are in that field, and then come up with some possible way in which good numerical linear algebra might help. Then tell someone else how a thing should really be done and get them to do the deed. The interesting person in question that led to this work was George Dantzig and the field was linear programming. My first contact with this interaction between Gene and George Dantzig must have been after Gene had the tableau method of the simplex method presented to him. I recall an incredulous statement from Gene roughly like, “It’s crazy; they use Gauss–Jordan elimination without pivot selection!” Gene proposed that some method of updating the LU decomposition of the basis matrix be found rather than the Gauss–Jordan update of the basis matrix inverse, as was used in the tableau. I needed a PhD thesis topic, so I became the “someone else” who was to do the deed. The updating process was Gene’s, and I was asked to see what I could say about its numerical stability using Wilkinson round-off analysis.
Methods For Modifying Matrix Factorizations (With P. E. Gill, W. Murray And M. A. Saunders)
Authors: Chan Raymond H , Greif Chen , O’Leary Dianne P
Published: November 2023

The paper describes methods for updating the Cholesky factorization of a symmetric positive-definite matrix, and the complete orthogonal factorization of a rectangular matrix. These methods were proposed as an alternative to updating the inverse and pseudo-inverse. The bulk of the paper was written at Stanford over the spring and summer of 1972. The joint paper was proposed by Gene during a visit (with Michael Saunders) to the National Physical Laboratory the previous year. My visit to Stanford was funded by Gene, and I will always be grateful for his support and encouragement during that summer. This was my first visit to the United States and it opened up many opportunities, including the chance to work with my subsequent colleagues Michael Saunders and Margaret Wright.
Commentary, By Walter Gautschi
Authors: Chan Raymond H , Greif Chen , O’Leary Dianne P
Published: November 2023

This group of five papers, especially the first and third, has a distinctly “interdisciplinary” character in the sense that classical analysis problems are recast in terms of, and successfully solved by, techniques of linear algebra and, vice versa, problems that have a linear algebra flavor are approached and solved using tools of classical analysis. A similar intriguing mix of analysis and algebra permeates the remaining three papers.
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.
The Accession of Lithuania to the EU
Authors: Auštrevičius Petras
Published: November 2023

Lithuania’s twentieth-century history has been shaped by a complicated domestic political life, on the one hand, and the long-standing relations of the country with the West, on the other. During the last century, Lithuania has not spent more than thirty-three years as an independent state.
Biosecurity and quarantine for preventing invasive species
Authors: Maynard Glynn , Nowell David
Published: November 2023

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

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.
Finite mixture models
Authors: Aitkin Murray , Francis Brian , Hinde John , Darnell Ross
Published: November 2023

The exponential family of distributions and its extensions discussed in earlier chapters are extremely useful in statistical analysis, but they cannot represent all types of data of scientific interest.
Congruences To Composite Moduli
Authors: Hardy G H , Wright E . M
Published: November 2023

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

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

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 >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

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

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 < k.
Democracy and Growth: A Case Study in Failed Causal Inference
Authors: Seawright Jason
Published: November 2023

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.
Decision science
Authors: Anderson Raymond
Published: November 2023

Man’s strength has always been his ability to change his environment to suit himself. He observes the ways things work, and controls disturbances to maintain consistent results, or experiments to improve them. In some instances the risks may be life threatening, but they can be lessened with proper controls. The greater the danger, the greater the need for protection! Business enterprises operate similarly, except the mechanisms involved are more formalised. Control can be viewed as having the following dimensions:
Measures of separation/ divergence
Authors: Anderson Raymond
Published: November 2023

These measures of goodness of fit have a fatal attraction. Although it is generally conceded among insiders that they do not mean a thing, high values are still a source of pride and satisfaction to their authors, however hard they may try to conceal these feelings.

J.S. Cramer (1987)

On a recent visit to Paris, my partner and I visited the Eiffel Tower, and opted to go to the top, level 3. This is 295 metres above the ground, and once at the top, I automatically felt slightly giddy, as though there were a slight sway in the tower. I had never felt this way before though, in spite of having looked over precipices of more than a kilometre on hiking trips in South Africa. When I mentioned the sway to my partner, she said, ‘Nonsense!’ I insisted that there must be some sway in the tower, whether because of wind (it was a calm day), the movement of lifts up and down its centre or legs, or the shifting of the many and gawking tourists on any of the three observation platforms. She conceded that there might be some sway, but thought it imperceptible, and insisted I was suffering from vertigo. I swore the contrary, but had no means of proving my point. No doubt it can be measured, but we had neither the tools nor access to information to settle the argument.
Information sharing
Authors: Anderson Raymond
Published: November 2023

In many competitive environments, people play with cards very close to their chests, and any evidence of collusion is considered cheating. In a game like credit however, there are risks that can cause all players to lose, which can be reduced if lenders show at least some of their cards. The real antagonist is not the competing lender, but the borrower; and the information being shared relates not to one’s own strategies, but the customers’. In explaining the emergence of the credit bureaux, and their ability to lower the cost of collecting and evaluating data, Barron and Staten (2003) state:

Lending markets almost always display an information asymmetry between borrowers and lenders. Borrowers typically have more accurate information than lenders about their likelihood of repaying a loan. Lenders have an obvious incentive to evaluate a borrower’s creditworthiness, and the outcome will affect whether to approve the loan, as well as its price. Borrowers have an incentive to signal their true risk (if it is low), or disguise it (if it is high). Given the amount of loan principal at stake, both parties have incentives to incur costs (often large ones) to reduce the information asymmetry.
Characteristic selection
Authors: Anderson Raymond
Published: November 2023

The product of the transformation process is a set of transformed variables, which are potential candidates for inclusion in a model. These may represent about 40 characteristics for a simple application scorecard, and increase to 200 or more if data is used from other internal sources (existing product holding and performance) and/or external sources (credit bureaux). For behavioural scorecards, there may be 60 or more for simple fixed-term repayment products and over 120 for transaction products. Obviously, the number has to be cut down; otherwise, the task can be overwhelming. This is especially true if: (i) numerous iterations are required to finalise a model; or (ii) the predictive modelling technique is calculation intensive. According to Thomas et al. (2002), the latter applies especially to logistic regression and decision trees. Thus, this is another instance where the complexity needs to be reduced. This chapter covers several aspects of characteristic selection, or reduction:
Scorecard calibration
Authors: Anderson Raymond
Published: November 2023

A scorecard has now been developed for each of the identified segments, but the job is not yet done. Like any measuring instrument (e.g. a weigh scale), scores must be calibrated, to give readings in the appropriate units. This ensures consistency of meaning across scorecards, and provides an opportunity to determine probability estimates for use in finance and elsewhere. It can be done not only as part of the scorecard development, but also at various points thereafter (recalibration). Calibration is required whenever the scores cannot be directly associated with the required probability estimates, whether as the result of: (i) the statistical modelling technique used; (ii) the passing of time, or (iii) differences between the model’s target variable and the metric to be estimated. The latter can arise from differences between the definitions (like good/bad versus default) or time frames (short-versus long-term).
Implementation
Authors: Anderson Raymond
Published: November 2023

How will the scorecards be applied? How will the results be monitored? How will the customer be advised? These questions should have been considered before the scorecard development started, but if not, the answers are now critical, and will involve varying degrees of technical complexity. This chapter covers some of the primary considerations, under the headings of:

When lenders use credit scoring for the first time, it brings about a profound change in the way they do business, to the extent that it influences the corporate culture and relationships with customers. This section covers several aspects relating to decision automation, and although they apply primarily to greenfield developments, some of them must also be considered for brownfields: