bluuebunny/crossref_metadata_2025_split huggingface dataset:
Bayesian Model Diagnostics Based on Artificial Autoregressive Errors*
Authors: Peruggia Mario 
Published: November 2023

Abstract
This article provides a synopsis of the material presented in Peruggia (2007), where I develop a simple and effective device for ascertaining the quality of the modeling choices and detecting lack-of-fit in hierarchical Bayes models. These models will typically specify regression equations for one or more parameters, with errors that are often assumed to be i.i.d. Gaussian. To build the diagnostic tool, I specify an artificial autoregressive structure (AAR) in the probability model for the errors that incorporates the i.i.d. model as a special case and assess lack-of-fit by examining the posterior distribution of the AAR parameters.
Some Other Theorems
Authors: Medvegyev Péter 
Published: November 2023

Abstract
In this chapter we shall discuss some further theorems from the general theory of stochastic processes. First we shall prove the so-called Doob–Meyer decomposition. By the Doob–Meyer decomposition every integrable submartingale is a semimartingale. We shall also prove the theorem of Bichteler and Dellacherie, which states that the semimartingales are the only ‘good integrators’.
Processes With Independent Increments
Authors: Medvegyev Péter 
Published: November 2023

Abstract
In this chapter we discuss the classical theory of processes with independent increments. In the first section we return to the theory of Lévy processes. The increments of Lévy processes are not only independent but they are also stationary. Lévy processes are semimartingales, but the same is not true for processes with independent increments. In the second part of the chapter we show the generalization of the Lévy–Khintchine formula to processes with just independent increments.
The Calabi Conjecture
Authors: Joyce Dominic D 
Published: November 2023

Abstract
The conjecture was posed by Calabi in 1954, who also showed that if g exists it must be unique. It was eventually proved by Yau in 1976, [344,345]. Before this, Aubin [15] had made significant progress towards a proof. In this chapter we will give a proof of the Calabi Conjecture that broadly follows Yau ‘s own proof, with some differences. My main references were Yau ‘s paper [345], and the treatment given in Aubin ‘s book [16, 7]. The proof is also explained, in French, by Bourguignon et al. [42].
Calabi–Yau Manifolds
Authors: Joyce Dominic D 
Published: November 2023

Abstract
Although much of the chapter is algebraic geometry, a rather technical subject, we will try to present it in an elementary way and keep the technicalities to a minimum. One thing we will not say much about is what the metrics on the Calabi–Yau manifolds are like: here we are mostly interested in the underlying complex manifolds. In fact little is known about general Calabi–Yau metrics. The subject of Calabi–Yau manifolds is too big to be properly described in one chapter of a book. We aim to cover three main areas in detail: firstly, the basic differential geometry of Calabi–Yau manifolds; secondly, orbifolds and crepant resolutions; and thirdly, ways of constructing Calabi–Yau manifolds. There are other important areas, such as rational curves on Calabi–Yau 3-folds, that we shall not have space to discuss.
Introduction: Challenges and Opportunities of the Enlargement of the European Union
Authors: Verheugen GÜnter 
Published: November 2023

Abstract
Politically enlargement became a reality on 1 May 2004, when eight new Member States of Central and southern Europe—Poland, the three Baltic states, the Czech Republic, Slovakia, Slovenia, Hungary—as well as Malta and Cyprus joined the EU. It would be premature to make a final judgement. However, the most important effects of the strategic decision on enlargement have already taken place: today’s Europe is a safer and better place to live in and to do business in than fifteen years ago.
The Danish Presidency: Conclusion of the Negotiations
Authors: Christoffersen Poul Skytte 
Published: November 2023

Abstract
The Swedish, Belgian, and Spanish presidencies had led to considerable progress in the enlargement negotiations. The road map established by the Commission at the end of 2000 had generally been adhered to. However, the Danish presidency, which started on 1 July 2002, had a huge task before it, if the ambition—to conclude the enlargement negotiations before the end of the year—should be crowned with success.
The acting brain
Authors: Rizzolatti Giacomo , Sinigaglia Corrado , Anderson Frances 
Published: November 2023

Abstract
In the first Chapter we have seen that the act of picking up an object, our cup of coffee for example, is a combination of two processes, reaching and grasping, which, though independent, are inter-coordinated. It is commonly thought that the former precedes the latter, but this is not so. Recordings of the movements of the arm and hand show that these processes start and proceed in parallel. The arm moves towards the cup and contemporaneously the hand assumes the shape necessary to grasp it.
Trees and Games of Logic
Authors: Higgins Peter M 
Published: November 2023

Abstract
Many games and puzzles, perhaps more than you realize, are tree searches. Examples include games like Mastermind and puzzles such as the modern creation known by the Japanese name of Doku, although this puzzle type seems first to have appeared in the US-based Dell magazine many years earlier. The most difficult are the board games such as checkers and chess.
The simplest is tic-tac-toe or noughts and crosses. This pastime is complicated enough to be a real game yet simple enough for we humans to master it completely.
The first player marks his cross in any of the nine squares while the second player counters with her noughts. They continue until one or other gets three of their symbols in a line (horizontal, vertical, or diagonal) or until all nine squares are occupied without this happening in which case the game is drawn. It is obviously a big advantage to go first in this game, yet, if the second player is careful, she can avoid defeat and even win if the first is careless.
The Nature of Nets
Authors: Higgins Peter M 
Published: November 2023

Abstract
In this chapter you will be introduced to an array of different kinds of questions that concern general networks that are not just trees but in which cycles and multiple edges are permitted between nodes. At first inspection the problems and queries may not seem to be about networks but the underlying network comes to life as a natural model of the situation in each case. But first we look a little more closely at the reasons why nets are becoming more noticed.
‘I’ ve danced with a man, who has danced with a girl, who has danced with the Prince of Wales!’ The idea of this was enough to send the girl singing this old song into raptures. However, as has often been observed, this kind of thing is bound to happen once we start to mix with prominent people. Even if we make a point of avoiding celebrities all our lives, it was observed, apparently by the Hungarian writer Frigyes Karinthy as early as 1929, that it is more than likely that any two people on Earth could be linked through a chain of no more than five personal acquaintances.
Colouring and Planarity
Authors: Higgins Peter M 
Published: November 2023

Abstract
This chapter begins and ends with questions that can be resolved through talking about colouring the vertices of networks while respecting certain rules. It is striking that a topic that sounds very technical lies at the heart of a variety of questions that vary from the colouring of maps, to the guarding of museums, to deep mathematical questions that are applicable particularly in economics. The common thread throughout these investigations is planarity of the networks but its development from these problems is surprising and in one instance the application of network ideas emerges very much out of the blue.
How to Traverse a Network
Authors: Higgins Peter M 
Published: November 2023

Abstract
Let us now return to Euler’s resolution of the question of finding traversing walks in networks. Our explanation is thoroughly modern and is not expressed in the fashion that Euler would have used. Although he is regarded as an excellent expositor of his own work and is renowned for producing mathematics that was free of error, he suffered greatly from the fact that network theory was a new and unrecognized field. It would have been strange in the extreme to devise the kind of language that we have introduced here, incorporating as it does a large number of interrelated ideas, simply in order to deal with one or two problem types. Lacking this language and notation was however a great handicap and Euler struggled to convey his new ideas, despite the fact that, to the modern mathematical outlook, they are not especially difficult to come to terms with and put into practice.
For Connoisseurs
Authors: Higgins Peter M 
Published: November 2023

Abstract
This final chapter is intended to give a little more mathematical explanation for those readers who would appreciate some without having to pursue another source. The level of difficulty will vary—much of it is not very hard at all but, unlike in the main part of the book, I will assume that the reader has some familiarity with mathematical ideas and notation. However, most readers will be able to gain from dipping into the text here and there.
In some places I will be using proof by induction. This is the mathematical technique where the proof is established by building from one case to the next. A case study that is relevant here is the fact that any tree with n nodes has n – 1 edges.
EU Social Policy, or, How Far Up Do You Like Your Safety Net?
Authors: Johnson Ailish 
Published: November 2023

Abstract
Social issues figured prominently during the debate surrounding the European constitution, and claims that the EU is not sufficiently concerned with social policy contributed to its collapse. The French ‘no’ vote in May 2005 was in part spurred by concerns about high rates of national unemployment and fears that increased integration would contribute to further job losses and lower social standards. The French media symbolized this fear in the mythical ‘Polish plumber’, a specter of cheap Eastern European labor come to rob the French, and perhaps all of western and northern Europe, of their well-paid and high- benefit jobs. And yet this backlash about the weakness of the EU’s role in social policy came at a time when EU governance of social policy had never been greater.
Dispersal
Authors: Davis Mark A 
Published: November 2023

Abstract
The invasion process has been described as a series of stages (Carlton 1985, Williamson and Fitter 1996, Richardson et al. 2000a, Kolar and Lodge 2001). Some have characterized three stages: arrival, establishment, and spread (Williamson 1996, Freckleton et al. 2006) or establishment, spread, and integration (Marchetti et al. 2004); some have proposed four stages: arrival, establishment, spread, and adjustment (Ricklefs 2005, Reise et al. 2006).
Understanding and predicting invasions: an Integrated Approach
Authors: Davis Mark A 
Published: November 2023

Abstract
Despite confronting a very complex system, in which history plays a major role, invasion biologists have relentlessly tried to make the field a predictive science. There are likely two reasons for these persistent efforts. First, if the science is going to be able to inform management efforts, the field needs to provide knowledge that involves some reliable cause and effect relationships, i.e. predictability.
Framing biological invasions
Authors: Davis Mark A 
Published: November 2023

Abstract
Although quantum physicists seemed to have recently questioned the veracity of realism (Gröblacher et al. 2007), and post-modernists argue that humans do not have any access to reality (Allen et al. 2005), most scientists believe, not only that an external reality exists but, that humans are able to access at least some of this reality through our senses and instruments (Ellis 2005).
Conclusion
Authors: Davis Mark A 
Published: November 2023

Abstract
Nature is continually changing, but it does not have a goal. Nor, is there only one valid nature that can or should exist at a particular place and point in time (Diamond 1987, Hull and Robertson 2000). As scientists who both study and value nature, we need to operate somewhere between passively accepting the inevitability of change and obsessively trying to preserve the world as we have known it.
Filtering Model And Kallianpur–Striebel Formula
Authors: Xiong Jie 
Published: November 2023

Abstract
In this chapter, we introduce the basic setup of the filtering models to be studied in this book. Then we demonstrate that the optimal filter is given by the conditional distribution of the signal. Bayes ‘ formula in the filtering setup, is called the Kallianpur–Striebel formula, which is the key in the development of non-linear filtering theory. We will derive this formula in Section 5.2. We establish the filtering equations in Section 5.3. Finally, in Section 5.4, we give a particle-system representation of the optimal filter that will be useful in understanding numerical schemes for the optimal filter.
Uniqueness Of The Solution For Zakai ‘S Equation
Authors: Xiong Jie 
Published: November 2023

Abstract
In this chapter, we prove the uniqueness of the solution to Zakai ‘s equation by transforming it to an SDE in a Hilbert space and by making use of estimates based on Hilbert-space techniques. Most of the material in Sections 6.2–6.4 is taken from Kurtz and Xiong [97] where a large class of linear stochastic partial differential equations are studied. Although the techniques we introduce here are for Zakai ‘s equation, they can be applied to other classes of linear SPDEs that includes Zakai ‘s equation as a special example.
Stability Of Non-Linear Filtering
Authors: Xiong Jie 
Published: November 2023

Abstract
In this chapter, we consider the stability of the non-linear filter for the case when the observation and signal noises are independent, i.e. c 0 in the filtering model introduced in the previous chapters. In this case, X t is a time-homogeneous Markov process taking values in Rd. We will consider the filtering problem with a state space that is more general than what we studied in the previous chapters, namely, we assume that X t is a continuous time-homogeneous Markov process in a Polish space S .
The Series Of Primes (1)
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
The positive integers. The positive integers form the primary subject-matter of arithmetic, but it is often essential to regard them as a subclass of the integers or of some larger class of numbers. In what follows the letters will usually denote integers, which will sometimes, but not always, be subject to further restrictions, such as to be positive or non-negative. We shall often use the word ‘number’ as meaning ‘integer’ (or ‘positive integer’, etc.), when it is clear from the context that we are considering only numbers of this particular class. An integer ais said to be divisibleby another integer b, not 0, if there is a third integer csuch that If aand bare positive, cis necessarily positive. We express the fact that a is divisible by b, or bis a divisorof a, by b|a.
The Series Of Primes (2)
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
First proof of Euclid’s second theorem. Euclid’s own proof of Theorem 4 was as follows. Let 2, 3, 5, pbe the aggregate of primes up to p, and let Then qis not divisible by any of the numbers 2, 3, 5,, p. It is therefore either prime, or divisible by a prime between pand q. In either case there is a prime greater than p, which proves the theorem. The theorem is equivalent to Further deductions from Euclid ’s argument. If pis the nth prime pn, and qis defined as in (2.1.1), it is plain that This inequality enables us to assign an upper limit to the rate of increase of pn, and a lower limit to that of π(x). We can, however, obtain better limits as follows.
Congruences And Residues
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
Highest common divisor and least common multiple. We have already defined the highest common divisor (a, b)of two numbers aand b. There is a simple formula for this number. We denote by min(x, y)and max(x, y)the lesser and the greater of xand y. Thus min(1, 2)= 1, max(l, 1)= 1. This theorem is an immediate consequence of Theorem 2 and the definition of (a, b). The least common multipleof two numbers aand bis the least positive number which is divisible by both aand b. We denote it by {a, b}, so that a|{a, b}, b|{a, b}, and {a, b} is the least number which has this property. denotes a product extended over all prime values of p. The symbol denotes a product extended over all primes which divide m. In the first formula of Theorem 50, αis zero unless p|a(so that the product is really a finite product). We might equally well write In this case every αwould be positive.
Fermat’s Theorem And Its Consequences
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
Fermat’s theorem. In this chapter we apply the general ideas of Ch. V to the proof of a series of classical theorems, due mainly to Fermat, Euler, Legendre, and Gauss. Theorem 71 (Fermat’s theorem). If p is prime, and pf a, then The congruences (6.1.1) and (6.1.2) are equivalent when pf a; and (6.1.1) is trivial when p a, since then ap0 a. Hence Theorems 70 and 71 are equivalent. Theorem 71 is a particular case of the more general Theorem 72 (The Fermat–Euler theorem). If (a, m)= 1, then If xruns through a complete system of residues prime to m, then, by Theorem 58, axalso runs through such a system. Hence, taking the product of each set, we have Since every number xis prime to m, their product is prime to m; and hence, by Theorem 55,
Continued Fractions
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
Finite continued fractions. We shall describe the function as a finite continued fraction,or, when there is no risk of ambiguity, simply as a continued fraction.Continued fractions are important in many branches of mathematics, and particularly in the theory of approximation to real numbers by rationals. There are more general types of continued fractions in which the ‘numerators’ are not all 1’s, but we shall not require them here. The formula (10.1.1) is cumbrous, and we shall usually write the continued fraction in one of the two forms There is a clash between our notation here and that of § 6.11, which we shall use again later in the chapter (for example in § 10.5). In § 6.11, [x] was defined as the integral part of x; while here [a0] means simply a0. The ambiguity should not confuse the reader, since we use [a0] here merely as a special case of [a0, a1, ..., an]. The square bracket in this sense will seldom occur with a single letter inside it, and will not then be important.
Some Diophantine Equations
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
Fermat’s last theorem. ‘Fermat’s last theorem’ asserts that the equation where nis an integer greater than 2, has no integral solutions, except the trivial solutions in which one of the variables is 0. The theorem has never been proved for all n,† or even in an infinity of genuinely distinct cases, but it is known to be true for 2 &lt; n &lt;619. In this chapter we shall be concerned only with the two simplest cases of the theorem, in which n3 and n4. The case n4 is easy, and the case n3 provides an excellent illustration of the use of the ideas of Ch. XII. The equation x2 y2 z2. The equation (13.1.1) is soluble when n2; the most familiar solutions are 3, 4, 5 and 5, 12, 13. We dispose of this problem first. It is plain that we may suppose x, y, zpositive, without loss of generality. Hence, if x, y, zis a solution with (x, y) d, then x dxt, y dyt, z dzt, and xt, yt, zt is a solution with (xt, yt) 1. We may therefore suppose that 1, the general solution being a multiple of a solution satisfying this condition. Finally which is impossible; so that one of xand ymust be odd and the other even. It is therefore sufficient for our purpose to prove the theorem which follows.
The Arithmetical Functions φ(n), μ(n), d(n), ς (n), r(n)
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
The function φ(n). In this and the next two chapters we shall study the properties of certain ‘arithmetical functions’ of n, that is to say functions f (n)of the positive integer ndefined in a manner which expresses some arithmetical property of n. The function φ(n)was defined in § 5.5, for n &gt;1, as the number of positive integers less than and prime to n. We proved (Theorem 62) that This formula is also an immediate consequence of the general principle expressed by the theorem which follows. If ais prime to b, then the number of integers not greater than n, and divisible by both aand b, is and aand dare uniquely determined by hand n. Conversely, every fraction a/d, for which d|nand (16.2.2) is satisfied, appears in the set (16.2.1), though in general not in reduced form.
The Series Of Primes (3)
Authors: Hardy G H , Wright E . M 
Published: November 2023

Abstract
The functions ϑ(x)and Ψ(x). In this chapter we return to the problems concerning the distribution of primes of which we gave a preliminary account in the first two chapters. There we proved nothing except Euclid’s Theorem 4 and the slight extensions contained in §§ 2.1–6. Here we develop the theory much further and, in particular, prove Theorem 6 (the Prime Number Theorem). We begin, however, by proving the much simpler Theorem 7. Our proof of Theorems 6 and 7 depends upon the properties of a function there being a contribution log 2 from 2, 4, and 8, and a contribution log 3 from 3 and 9.If pmis the highest power of pnot exceeding x, log poccurs mtimes in Ψ(x).Also pmis the highest power of pwhich divides any number up to x, so that where U(x)is the least common multiple of all numbers up to x. We can also express Ψ(x)in the form † Throughout this chapter x(and yand t) are not necessarily integral. On the other hand, m, n, h, k, etc., are positive integers and p, as usual, is a prime. We suppose always that x): 1.
Odds and ends
Authors: Anderson Raymond 
Published: November 2023

Abstract
This section is used for topics that do not fit neatly elsewhere in this module. These are treated under three headings: descriptive statistics—covering cluster analysis and factor analysis; forecasting—covering Markov chains and survival analysis; and concepts—which include correlations, interactions, monotonicity, and normalization.
Credit scoring is almost exclusively associated with predictive modelling, but there are other techniques that are sometimes encountered. Descriptive techniques are used to describe the data, and are sometimes used as part of the scorecard development process, either to gain a better understanding of the data, or to create new variables. In contrast, forecasting techniques are used to do modelling at the portfolio level, to aid in various finance functions. Both categories each have two main techniques associated with them, as shown in Table 9.1, which are covered next.
Minds and machines
Authors: Anderson Raymond 
Published: November 2023

Abstract
A lot of business applications, such as financial modelling, are actually scientific in nature and even though the nature of the problems may be different in business, the mathematics is very similar to tasks we have already tackled in science.
Chorafas (1990)
When the above is read, one must understand science as also including fields like economics, engineering, psychology, and others. It is also not only the maths that is similar, but also much relating to the people and computers that are employed. By extension, this also applies to credit scoring. While it tries to bring science into lending decisions, it will always be both science and art. Creativity can be used at all stages in the process, but some developers wield their paintbrushes better than others. This section focuses on both the people and the software: