bluuebunny/crossref_metadata_2025_split huggingface dataset:
The basic coalescent
Authors: Hein Jotun , Schierup Mikkel H , Wiuf Carsten 
Published: November 2023

Abstract
In this chapter, we first motivate the need for a mathematical model that can describe the process generating genetic data, with specific emphasis on human variation. We assume that genetic data are in the form of DNA sequence data. The sequences or genes are all homologous copies of the same genetic region in the genome of a species. Whether one or both copies of a gene in an individual is sampled does not matter here, what matters is their number and genetic type. Such data are collected from one or several present-day populations of a single species, and from this sample we want to infer details about the evolutionary processes that created the data. (A population is here best understood as a population of genes, rather than a population of individuals, because we focus at the level of genes.) The inferential analysis is retrospective; we seek to understand aspects of the sample ‘s (and the population ‘s) evolutionary past through analysis of the present day sample. Below we sketch an analysis of a data set from humans to make more clear the different types of questions that coalescent theory seeks to answer.
The human genome consists of twenty-two autosomal chromosomes, the two sex chromosomes X and Y, and the mitochondrion. One representative sequence of the human genome has recently been approximately determined, and is presently subject to refinement. Additionally, a major effort has now been directed towards determining the population
General introduction
Authors: Coolen A C C , Kühn R , Sollich P 
Published: November 2023

Abstract
1.1 Principles of neural information processing
The brain is a piece of hardware that performs quite sophisticated information processing tasks, using microscopic elements and operations which are fundamentally different from the ones on which present-day computers are based. The microscopic processors in the brain, the nerve cells or neurons (see Figure 1.1), are excitable brain cells which can be triggered to produce electrical pulses (spikes), by which they communicate with neighbouring cells. These neurons are rather noisy elements, which operate in parallel. They do not execute a fixed ‘program’ on a given set of ‘data’, but they communicate signals through relay stations (the synapses), located at the junctions where the output channel (axon) of one neuron meets an input channel (dendrite) or the cell body of another. The strengths of these relay stations, or synaptic efficaices, are continuously being updated, albeit slowly. The neurons of each given brain region are organized and wired in a specific network, the structure of which can vary from very regular (especially in regions responsible for pre-processing of sensory data) to almost amorphous (especially in the ‘higher’ regions of the brain, where cognitive functions are performed), see Figure 1.2. The dynamic relay stations, or synapses, in combination with some adjustable intrinsic neuron properties, represent both ‘data’ and ‘program’ of the network. Hence program and data change all the time.
Layered networks
Authors: Coolen A C C , Kühn R , Sollich P 
Published: November 2023

Abstract
All elementary logical operations that we encountered in the previous chapter could not only be realized with McCulloch–Pitts neurons, but even with a single, McCulloch–Pitts neuron. The question naturally arises whether all operations {0, 1} N → {0, 1} can be performed with a single McCulloch–Pitts neuron.
Dynamics of online learning in binary perceptrons
Authors: Coolen A C C , Kühn R , Sollich P 
Published: November 2023

Abstract
17.1 Probabilistic definitions, performance measures
In this chapter we study the dynamics of supervised learning in artificial feedforward neural networks. The basic scenario is identical to that introduced in Part I: a student neural network executes a certain operation S: Ω → R, which is parametrized by a vector J , usually representing synaptic weights and/or neuronal thresholds. Here Ω ⊆ IRN denotes the set of all possible questions and R denotes the set of all possible answers. The student is being trained to emulate a given teacher, which executes some as yet unknown operation T : Ω → R. In order to achieve the objective, the student network S tries gradually to improve its performance by adapting its parameters J according to an iterative procedure, using only input vectors (or questions) ξ which are drawn at random from a fixed training set D ⊆ Ω of size | D ,| and the corresponding values of the teacher outputs T (ξ) (the correct answers). The iterative procedure, or learning rule, is not allowed to involve any further knowledge of the operation T . As far as the student is concerned the teacher is an oracle, or black box. The only information available about the inner workings of the black box is contained in the various answers T (ξ) it provides; see Figure 17.1. For simplicity we assume each question ξto be equally likely to occur; generalization of what follows to the case where the question probabilities are non-uniform is straightforward.
Traces of Intertwiners for 𝔘q(g)
Authors: Etingof Pavel , Latour Frédéric 
Published: November 2023

Abstract
Definition 7.1 Given a finite-dimensional representation W of 𝔘q (𝔤), we will write R(λ) = R(−λ − ρ) and we will define the generalized Macdonald– Ruijsenaars operator DW : Fun(h*, V [0]) → Fun(h*, V [0]) by the formula.
The provenance of the astrolabe collection at the national maritime museum
Authors: Cleempoel Koenraad Van 
Published: November 2023

Abstract
The National Maritime Museum in Greenwich belongs to a select group of institutions that house large collections of planispheric astrolabes. Only the collection at the Museum of the History of Science in Oxford is larger, while the Adler Planetarium in Chicago, the Istituto e Museo diStoria della Scienza in Florence, and the British Museum in London have comparable numbers of astrolabes in their collections. In terms of quality, however, the collection at the National Maritime Museum presents perhaps the finest balance between astrolabes of the Western and Eastern tradition.
Representation Theorems
Authors: Dunn J Michael , Hardegree Gary M 
Published: November 2023

Abstract
We annunciated a theme in Remark 6.5.3 which we want to reemphasize here. Propositions can be understood as sets of “possible worlds “ or, as we now stress, more generally as sets of “information states. “ The latter is more general in that there can be states of information that are inconsistent, incomplete, or both, and so do not correspond to possible worlds. An “information frame “ will always consist of at least a set U whose elements are regarded as “states of information. “ It may have additional features,
A Study of Logics
Authors: John P. Cleave 
Published: November 2023

Abstract
It is a fact of modern scientific thought that there is an enormous variety of logical systems - such as classical logic, intuitionist logic, temporal logic, and Hoare logic, to name but a few - which have originated in the areas of mathematical logic and computer science. In this book the author presents a systematic study of this rich harvest of logics via Tarski's well-known axiomatization of the notion of logical consequence. Those with a modest acquaintance of modern formal logic will find this to be a readable and not too technical account which will demonstrate the current diversity and profusion of logics. In particular, undergraduate and postgraduate students in mathematics, philosophy, computer science, and artificial intelligence will enjoy this introductory survey to the field.
Topological transformation groups
Authors: Kawakubo Katsuo 
Published: November 2023

Abstract
In this chapter, we discuss the basic notions of topological transformation groups, such as topological groups, fixed point sets, orbits, orbit spaces, orbit types, isotropy groups, and induced transformation groups. The discussion takes place from the point of view of general topology. These ideas display their real power in the study of smooth transformation groups on manifolds, which we shall deal with later.
Graphs
Authors: Read Ronald C , Wilson Robin J 
Published: November 2023

Abstract
A graph G = (V, E) consists of a non-empty finite set V of elements, called vertices, and a set E of pairs of distinct elements of V, called edges. Any graph can be represented as a diagram, with vertices represented by points and edges represented by lines joining the corresponding pairs of points. The following diagram depicts a graph with seven vertices and nine edges.
Digraphs
Authors: Read Ronald C , Wilson Robin J 
Published: November 2023

Abstract
The underlying graph of a digraph D is the graph obtained by replacing each arc of D by the corresponding undirected edge. The converse of D is the digraph obtained by reversing the direction of each arc in D, and the complement of D is the digraph with vertex set Vin which vw is an arc if and only if it is not an arc in D. A digraph is self-converse if it is isomorphic to its converse, and self-complementary if it is isomorphic to its complement., A tournament is a digraph in which each pair of vertices is joined by exactly one arc.
Second-Order Partial Differential Equations
Authors: Collins Peter J 
Published: November 2023

Abstract
This chapter discusses the equation where p, q, r, s, t denote the partial derivatives of the twice continuously differentiable real-valued function z = z(x, y) (which implies z xy = z yx) and A, B, C are real-valued functions of (x, y) in an appropriate domain in ℝ2. The function φ is a continuous function of the ‘lower order’ (and necessarily continuous) functions z, p, q, as well as of the variables x, y. Note immediately that A, B, C are not functions of z (unlike the functions P, Q, R of Chapter 5). An equation such as (1), where the terms involving the highest-order derivatives are linear in those derivatives (as here, on the left-hand side of (1)), is said to have linear principal part. Equation (1) is linear if the function φ is given in the form where D, E, F, G are functions of x and y. By an appropriate change of independent variables equation (1) in z = z(x, y) can be transformed to an equation in Z = Z(ξ, η) in canonical form, where In fact, there are three posssible canonical forms, according to when the functions A, B, C of (x, y) satisfy B 2 &lt; AC, B 2 &gt; AC or B 2 = AC. In this way, one classifies the partial differential equation (1). Details of this classification now follow.
The Fredholm Alternative
Authors: Collins Peter J 
Published: November 2023

Abstract
The Fredholm Alternative analyses the solutions of the non-homogeneous Fredholm equation where f : [a, b] → ℝ and K : [a, b]2 → ℝ are continuous and λ is a constant. We shall only establish it in the special and useful case when K = K(x, t) is degenerate (some authors say, of finite rank), that is, when and g j : [a, b] → ℝ, h j : [a, b] → ℝ are continuous (j = 1, ..., n). Note that the summation is a finite one. As with Green's functions in Chapter 4, the method of proof we shall employ also gives a practical method for solving problems. In order to motivate both the statement and proof of the Fredholm Alternative Theorem, we now discuss an equation with perhaps the simplest degenerate kernel K = K(x, t) of any interest; namely, where g and h are continuous on [a, b] and neither of which is identically zero on [a, b].
Transform Methods
Authors: Collins Peter J 
Published: November 2023

Abstract
This chapter introduces a powerful technique for solving differential equations, both ordinary and partial, and integral equations, involving methods quite different in nature from those discussed earlier. The technique involves applying an integral operator 𝒯 = 𝒯x→v to ‘transform’ a function y = y(x) of the independent real variable x to a new function 𝒯 (y) = 𝒯 (y)(v) of the real or complex variable v. If y solves, say, a differential equation (DE) – and satisfies conditions allowing the integral operator to be applied – the function 𝒯 (y) will be found to solve another equation (𝒯DE), perhaps also (but not necessarily) a differential equation. This gives us a new method for solving a differential equation (DE): We will find that, in solving differential equations, there can be at most one function y corresponding to a transformed function 𝒯(y), and hence that we will have a one-to-one correspondence between transformable y's solving (DE) and transformed functions 𝒯(y)’s solving (𝒯DE). We should note that, as with the Green’s function method discussed in section 4.2, the method can allow us to build in initial or boundary conditions from the start and hence produce (following our discussion in the last paragraph) the unique solution of a problem consisting of a differential equation and side conditions.
The Numerical Methods Programming Projects Book
Authors: Grandine Thomas A 
Published: November 2023

Abstract
Traditional numerical analysis books concentrate either on the mathematical or programming aspects of numerical algorithms. This textbook is different inasmuch as it emphasizes the relevance of these techniques to the real world and the use of a widely available library of numerical software in their application. The book consists of 22 carefully graded projects which will lead the reader through the techniques typically taught as part of a first course in numerical analysis. Throughout, readers are presented with projects which reflect very real problems that occur in science and industry. At the same time, they become accustomed to using a good library of numerical software when writing their programs. It is a theme of this book that the use of a solid, robust and bug-free software library will improve computational results and minimize the effort of programming. By integrating the use of the NAG (Numerical Algorithms Group) FORTRAN library into the projects, students will develop experience and expertise in the use of a software library and, by practical example, be better prepared for working further with numerical analysis libraries. This lively and entertaining text will provide a valuable complement to more traditional numerical analysis books. Answers to exercises are included as well as full documentation of the relevant library routines used.
Project 13
Authors: Grandine Thomas A 
Published: November 2023

Abstract
A fresh supply of crude melange has just been delivered from the planet Arrakis. Melange is a valuable spice with fantastic properties; it is known to slow the ageing process by a factor of nearly four, and those who take it in heavy doses are sometimes capable of seeing the future. In any case, the Production of Chemical Brilliance division of ACME has a contract with the Nutritional Union of the True Monarchy of Extraterrestrial Galaxies to refine the crude melange into a more pure form. To do this, the PCB division has designed a machine for NUTMEG which does the refining. This machine has a hopper which is initially loaded with 100 kg of crude spice. The machine is turned on and allowed to run. It starts producing pure melange at the rate of 1 kg/minute. However, this rate begins to fall off quickly, for the concentration of spice in the hopper decreases as more and more pure melange is removed. Suppose that the rate at which pure spice comes out of the machine is given as a function of time, say R(t).
Rings
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
We must start by defining our terms. It is all very well to talk vaguely about generalized number systems when trying to give a non-technical feeling for the objects which will be studied in this book, but in order to make things mathematically respectable we need to pin down precisely what we are going to mean by such phrases as ‘well-behaved addition’. What we are about to do when defining a ring is the analogue of setting out the rules for a game; it is necessary to have clearly formulated rules in order to know precisely what is allowed and what is not (and to settle disputes), and initially the rules may look very complicated; but with experience we get to know what is or is not permitted, and the rule-book can be put away. Similarly, the definition of a ring (which is a formalization of the idea of a generalized number system) looks complicated, but with practice we learn how to work with rings and can put aside the precise definition. So, do not be put off by the length of the definition. Also, it may seem that in this chapter we are flitting from topic to topic in a rather superficial way; this is so, but the topics we introduce briefly now will be given much more thorough development later in the book.
Field of quotients of an integral domain
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
Let R be an integral domain. We shall show that there is a field F containing R as a subring such that every element of F is a fraction formed from the elements of R. This generalizes the way in which every element of the field Q is a fraction of elements of 71... The method which we shall give for constructing F from R is rather complicated, but the important point is that it always works. Also, in many cases, given a particular integral domain R, it is easy to find or describe the corresponding field Fas will be seen from examples later. In Chapter 8 we shall need the existence of such a field F when R is an arbitrary UFD. The construction given below is motivated by the following consideration. The rational number 2/3 can be constructed as a pair (2, 3). However, we have to arrange things so that the pairs (2, 3) and (10, 15) give the same fraction since the fractions 2/3 and 10/ 15 are equal.
Fields and field extensions
Authors: Chatters A W , Hajarnavis C R 
Published: November 2023

Abstract
We shall now go rather more deeply into the theory of fields. In many ways a field is the best sort of system to work with, because it is always possible to divide by a non-zero element. The material of this chapter has many uses in the remainder of the book. In particular it is crucial to the study of roots of polynomial equations. This is of great importance in such areas as number theory and geometry.
Regular and Cartesia’s Closed Categories
Authors: Johnstone Peter T 
Published: November 2023

Abstract
In this book, we start from the assumption that the reader is already familiar with the basics of category theory: specifically, we assume that he has already seen the definitions of the terms category, functor, natural transformation, adjunction, limit and monad. These definitions can, of course, be found in any of the standard textbooks on category theory (for example, [742], [381] or [147]). However, there are a number of conventions on which the standard texts differ, and we need to state our position on these before proceeding further.
Indexed Categories And Fibrations
Authors: Johnstone Peter T 
Published: November 2023

Abstract
For many of the topics we wish to study in topos theory, it is impossible to ignore the fact that toposes and geometric morphisms form not just a category but a 2-category; that is, we have to take account also of the geometric transformations between geometric morphisms, as introduced in A4.1.1(b). Even amongst experienced category-theorists, there is often a good deal of reluctance to take 2-categorical (or, still worse, bicategorical) notions seriously; so the purpose of this section is to review as much of the basic theory as we shall need for subsequent developments.
Newton’s laws
Authors: Lunn Mary 
Published: November 2023

Abstract
Prior to the late nineteenth and early twentieth centuries the behaviour of the physical world was thought to be completely explained by the model proposed by Newton. In the sixteenth century Tycho Brahe made detailed and accurate astronomical observations which Johannes Kepler was able to analyse. Kepler’s laws are in fact a mathematical description of the motion of the planets. They are: 1. The path of each planet is an ellipse with the Sun at a focus.
Central forces
Authors: Lunn Mary 
Published: November 2023

Abstract
In this chapter we consider the application of the Newtonian model to the physical system which originally inspired its formulation–the Sun and its planets. On the basis of experimental evidence we write the attractive force between two gravitating bodies such as the Earth and the Sun, as along the line joining the two bodies, where M, m are the respective masses, r is the distance between them, and G is the universal gravitational constant (6.67 x 10"11 in SI units).
Impulsive forces
Authors: Lunn Mary 
Published: November 2023

Abstract
In the previous chapters we have dealt with continuously applied forces We have not yet tackled the behaviour of mechanical systems which eithe1 suffer some sort of collision or which are subjected to a ‘short sharp’ blow How do we describe mathematically the instantaneous effect of a blow from a hammer on a crowbar or the initial motion of a snooker ball after being struck by a cue? In both instances the time span of the blow (contact time) is so short as to be negligible, just a fraction of a second. In orde1 to extend our model we must first state what we mean by an impulsiv&lt; force.
Superstable theories
Authors: Pillay Anand 
Published: November 2023

Abstract
In this final chapter we give some scattered results on superstable theories. One of the main lines here is the development of some technology which helps towards generalizing results from the finite rank context to the general superstable context. A key point is the interaction between the local theory developed in Chapter 7 and Shelah’s ‘infinity rank’, R ∞. This will yield in some situations ‘continuity and definability’ of p-weight. In any case one of the goals of this chapter is to suitably generalize Buechler’s dichotomy theorem to regular types in superstable theories. This is done in section 5, under the assumption that T is countable and has NDOP and NOTOP (two structure properties); this result has great importance for some fine questions concerning the spectrum function. Note we also give some other versions of the dichotomy theorem, which do not depend on assuming NDOP and NOTOP, and which are relevant to Vaught’s conjecture for superstable theories. In the process of developing the technology, we mention various related ‘non-geometric’ results, such as the existence of regular types between arbitrary models, and definability questions in non-multidimensional theories.
Available Software
Authors: Dierckx Paul 
Published: November 2023

Abstract
In the preceding chapters we have regularly referred to software implementations of the smoothing spline algorithms. The most important of these routines haw been collected into a large FORTRAN subroutine package, called FITPACK (Dierckx 1987a, 1989). In this chapter we will give some details about its contents, organization and documentation. Finally, to conclude this book, we will review some of the important features of the proposed spline algorithms and indicate a number of possible directions for further research.
FITPACK is a collection of 83 routines, all written in standard FORTRAN and carefully checked with the PFORT-verifier (Ryder 1974) of Bell Labs. The package has been successfully implemented on a wide range of computers, from main frame to mini-computers and PCs. It contains no machine¬ dependent constants. Communication with the package is solely through the parameter list of one of the following 29 routines, classified as: