bluuebunny/crossref_metadata_2025 huggingface dataset:
Greeks and ‘origins’
Authors: Hodgkin Luke
Published: November 2023

Socrates: Then as between the calculating and measurement employed in building or commerce and the geometry and calculation employed in philosophy—well, should we say there is one sort of each, or should we recognize two sorts? Protarchus: On the strength of what has been said, I should give my vote for there being two.
Chinese mathematics
Authors: Hodgkin Luke
Published: November 2023

Fu Xi created the eight trigrams in remote antiquity to communicate the virtues of the gods and parallel the trend of events in earthly matters, [and he] invented the nine-nines algorithm to coordinate the variations in the hexagrams. (Liu Hui, cited Shen et al. 1999, p. 52 Mathematics is an important subject in the six arts. Through the ages all scholars who have participated in discussions on astronomy and calendars have to master it. However, you may consider it as a minor occupation, not as a major one.
Some preliminaries on measure theory
Authors: Ambrosio Luigi , Tilli Paolo
Published: November 2023

In this chapter we recall some preliminary results on abstract measure theory (see for instance Rudin 50], Simon 54], Evans and Gariepy 17], and Ambrosio et al 6]).
Hausdorff measures and covering theorems in metric Spaces
Authors: Ambrosio Luigi , Tilli Paolo
Published: November 2023

In this chapter we introduce and study the main properties of Hausdorff measures ℋk in a general metric setting. This class of measures provides a very general extension of the classical concepts of length, surface area, and volume. We also study some covering theorems in metric spaces and we apply them to compare, using the spherical densities, Hausdorff measures with Radon measures.
The basic coalescent
Authors: Hein Jotun , Schierup Mikkel H , Wiuf Carsten
Published: November 2023

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

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

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

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

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

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.
The construction and use of the astrolabe
Authors: Proctor David
Published: November 2023

Since earliest times, man has earnestly sought an under- standing of the stars, the planets, and especially of the Sun. The mysterious movements of these bodies and their real and supposed influence on the daily life of men have stirred minds to seek answers, build philosophies, and to make observations of measurement.
Representation Theorems
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

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,
Introduction
Authors: Diestel Reinhard
Published: November 2023

In his classic paper Über eine Eigenschaft der ebenen Komplexe, Wagner (1937a) tackles the following problem. Kuratowski’s theorem, in its excluded minor version, states that a finite graph is planar if and only if it has no minor isomorphic to K or to K.
A Study of Logics
Authors: John P. Cleave
Published: November 2023

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

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

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

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.
Problems based on famous topics in the History of Mathematics
Authors: Gofman Judita
Published: November 2023

This chapter presents problems treated by eminent mathematicians in the past. Our selection aims to provide readers with additional information on topics encountered in the school syllabus and to raise interest in the History of Mathematics (see recommended reading).
Second-Order Partial Differential Equations
Authors: Collins Peter J
Published: November 2023

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 < AC, B  2 > 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

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

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

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

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).
Adjunctions
Authors: McLarty Colin
Published: November 2023

Most of this chapter is devoted to proving fundamental results on adjunctions. More specific results and calculations are given in the exercises.
Rings
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

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

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

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

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

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

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

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

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< force.
Superstable theories
Authors: Pillay Anand
Published: November 2023

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

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:
Reduce as an algebraic programming language
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

A Reduce “program” is not a well-defined object like a Pascal or FORTRAN program, but is just a body of code intended to perform a particular task. It will usually be a collection of procedures (equivalent to procedures, functions and subroutines in other languages), control structures, let-rules and variable assignments. Often these will be read or loaded from a file, and then the “program” will be run by invoking a procedure, which may itself call other procedures. Alternatively, the “program” may run as it is being read just as it would if the statements were entered one-by-one from the keyboard.
Programming 1n (R)Lisp
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

In this very brief introduction to Lisp, we will try to avoid features very specific to CULisp (Cambridge Lisp), which we ourselves use, but we will sometimes mention differences between Lisps. There are many good (and some bad) books on Lisp: one we have found useful is Winston and Horn (1988) though this is based on Common Lisp (older editions used Maclisp). Cambridge Lisp’s built in commands are in lower case, though names in either case are allowed (and there is in fact an upper case version also). However, we will follow our usual conventions and put Lisp commands in upper case here for clarity.
Advanced topics
Authors: MacCallum Advanced topics [CT] Malcolm A H , Wright Francis J
Published: November 2023

The first part of this chapter builds on chapters 5 and 6 to intro duce algebraic use of symbolic mode via a case study in the development of a simple new Reduce facility. It then discusses the use of matrices as local rather than global objects and uses some symbolic procedures to provide complete (but inelegant) support for local matrices. The second part presents an example of using Reduce to generate FORTRAN 77 code for use in numerical computations, in which a Chebyshev-economized polynomial approximation is generated in Horner or nested form suitable for efficient numerical evaluation. Finally, a few remaining facilities are listed that have not been discussed in detail.
Differentiation: The Amplitwist Concept
Authors: Needham Tristan
Published: November 2023

To know the graph of an ordinary real function is to know the function completely, and so to understand curves is to understand real functions. The key insight of differential calculus is that if we take a common or garden curve, place it under a microscope and examine it using lenses of greater and greater magnifying power, each little piece looks like a straight line.When produced, these infinitesimal pieces of straight line are the tangents to the curve, and their directions describe the local behaviour of the curve. Thinking of the curve as the graph of f(x), these directions are in tum described by the derivative, f′ (x).Despite the fact that we cannot draw the graph of a complex function, in this chapter we shall see how it is still possible to describe the local behaviour of a complex mapping by means of a complex analogue of the ordinary derivative—the “amplitwist”.
Flows and Harmonic Functions
Authors: Needham Tristan
Published: November 2023

Later we shall see that the concept of a dual flow is very useful. For example, having found the flow of a fluid round an obstacle, the dual flow represents the electric field which solves an analogous problem in electrostatics.
Diagonalization and Self-Reference
Authors: Smullyan Raymond M
Published: November 2023

The main purpose of this book is to present a unified treatment of fixed points as they occur in Gödel's incompleteness proofs, recursion theory, combinatory logic, semantics, and metamathematics. The book provides a survey of introductory material and a summary of recent research. The first chapters are of an introductory nature and consist mainly of exercises with solutions given to most of them.
Polygons and area
Authors: Huxley M N
Published: November 2023

Given a simple closed curve, what is the area inside it? This is the problem of quadrature, important in the history of mathematics. At first to find the area meant to construct a square of equal area; later it meant to find a real number, the ratio of the area of the curve to the area of a unit square. To find a real number can mean one of four things.
Analysis on Symmetric Cones
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

The present book is the first to treat analysis on symmetric cones in a systematic way. It starts by describing, with the simplest available proofs, the Jordan algebra approach to the geometric and algebraic foundations of the theory due to M. Koecher and his school. In subsequent parts it discusses harmonic analysis and special functions associated to symmetric cones; it also tries these results together with the study of holomorphic functions on bounded symmetric domains of tube type. It contains a number of new results and new proofs of old results.
The Gamma Function Of A Symmetric Cone
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

This chapter begins the serious study of analysis on symmetric cones. In terms of the generalized determinant and minors we introduce Δs, a kind of generalized power function. With the aid of this we define and evaluate the generalized gamma and beta functions associated with the cone. This then leads to identities involving the Laplace transformation of the generalized power function and to differential identities. In Sections 2 and 3 we then discuss the theory of the analytic continuation of the generalized Riesz integrals and some of their applications.
Complex Jordan Algebras
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

In later chapters we will study complex tube domains over symmetric cones. These are domains in the complexifications of Euclidean Jordan algebras. The first two sections of the present chapter discuss the relevant definitions and basic facts, including some properties of the structure group of a complexified Euclidean Jordan algebra which will be needed later. The subsequent sections will not be needed later in this book; however, they contain basic background information. Section 3 gives a generalization of the Jordan canonical form of matrices, which is valid in any complex Jordan algebra. In Section 4 we prove that the class of Euclidean Jordan algebras coincides with the class of formally real Jordan algebras. In Section 5 we prove that the complexifications of Euclidean Jordan algebras exhaust the seemingly larger class of complex semi-simple Jordan algebras.
Water Waves: Relating Modern Theory to Advanced Engineering Applications
Authors: Rahman Matiur
Published: November 2023

The purpose of this book is to present a self contained introduction to the mathematical and physical aspects of the theory of water waves. The book is aimed at undergraduate and graduate levels for engineers, physical scientists and mathematicians. Each chapter is concluded with practical problems expressed as exercises and accompanied by ample references for further studies. The book consists of ten chapters arranged into three parts: Part I: Basic Fluid Mechanics and Solutions Techniques which cover chapters 1-3. Part II: Water Waves covering chapters 4-7. Part III Advanced Water Waves which covers chapters 8-10. Parts I and II are elementary in nature; whereas Part III is more advanced. The first three chapters give the derivations of the fundamental mathematical equations. Chapter 2 outlines appropriate differential equations to describe the physical phenomena, and Chapter 3 reviews solution techniques of some simplified partial differential equations. Chapter 4 gives the developmental of wave equations, including the essential boundary conditons and describes small amplitude wave theory. Chapter 5 deals with finite amplitude wave theory and Chapter 6 outlines the study of tidal dynamics in shallow water. For random wave case, the deterministic methods described in previous chapters do not hold good. Therefore, chapter 7 is clearly devoted to wave statistics and wave energy spectrum. The application of wave theory is demonstrated in Chapter 8. Chapter 9 examines the nonlinear long waves in shallow water from a mathematical view point. The book concludes with Chapter 10 which illustrates the inverse scattering technique to solve solitary wave problem.
Conjectures and solutions
Authors: Rahman Q I , Schmeisser G
Published: November 2023

Counting zeros and critical points as often as their multiplicities, we can say that a polynomial with n zeros has exactly n = 1 critical points. There is no such simple relation between the number of di.’Jtinct zeros of a polynomial and that of its derivative.
Computational Methods in Linear Prediction for Band-Limited Signals Based on Past Samples
Authors: Mugler D H
Published: November 2023

An accurate prediction of the “next” value of a signal from its past samples is an important problem for many applications. Linear prediction coding, (Elias 1955; Rabiner and Schafer 1993), even predicts values several steps ahead of the last data sample. For the transmission of a block of signal samples, the LPC method is to record the first several samples and then use a linear prediction formula to estimate the values of the rest of the samples in the block (Meyer 1994, p. 33).
Almost Sure Sampling Restoration of Band-Limited Stochastic Signals
Authors: Pogany T
Published: November 2023

The origin of sampling principles for stochastic signals is not exactly known. Work of Kotel’nikov (1933) in this area was first made known to Western mathematicians by Kolmogorov, who mentioned it in a brief manner at the end of his talk at the Symposium on Information Theory at MIT, held on September 10−12, 1956 (Kolmogorov 1956). This involves the well-known sampling series, usually attributed to Whittaker, Kotel’nikov and Shannon in the Western literature, but often caned the Kotel’nikov formula in Eastern Europe and the former Soviet Union countries.
The Structure of Groups of Prime Power Order
Authors: Leedharn-Green C R , McKay S
Published: November 2023

The coclass project (1980-1994) provided a new and powerful way to classify finite p-groups. This monograph gives a coherent account of the thinking from which the philosophy that lead to this classification developed. The authors provide a careful summary and explanation of the many and difficult original research papers on the coclass conjecture and the structure theorem, thus elucidating the background research for those new to the area as well as for experienced researchers. The classification philosophy has lead to many new and challenging problems. Amongst those considered are problems about pro-p-groups, the Grigorchuk group, the Nottingham group as well as linear pro-p-groups. Throughout the book the authors have used a wide range of algebraic techniques and have developed from first principles, or from basic and well known results, the cohomology of groups, spectral sequences, and representation theory. This comprehensive and long-awaited survey of the recent and current research on the structure of finite p-groups will be an important reference for all group theorists.
Introduction And Prerequisites
Authors: Henstock Ralph
Published: November 2023

The idea behind this book’s theory can be approached from the calculus definition of the integral inverse of the derivative, thus stepping directly from the mathematics of Newton and Leibnitz into the post-Lebesgue modern mathematics.
Biordered sets
Authors: Higgins Peter M
Published: November 2023

In Section 1.5 the notion of the biordered set of a semigroupS was introduced, along with the associated idea of a sandwich set of an ordered pair of idempotents,S(e,f). That an element may always be drawn fromS(e,f) in caseS is regular was used implicitly in Theorem 1.4.17, and sandwich sets were also used to describe those regular semigroups in which the natural partial order is compatible with multiplication (Theorem 1.5.10 and Corollary 1.5.11).
Initial concepts and first results
Authors: Helemskii A YA
Published: November 2023

The subject of this book is a kind of synthesis. In the well-known textbook by Rickart it is said jokingly, but not without justification, that the head of the theory is in algebra and the feet in analysis [64, vi]. To characterize the present chapter such a comparison could be extended. Here our subject is presented as a new-born baby and so it has a disproportionately large head and small feet: in fact the reader will see that at the beginning of the account algebra is somewhat dominant compared with analysis. In the second chapter — as is customary with the growth of a body — the proportion begins to even out and there appear the first significant results based on an equally balanced interaction between the two “basic” disciplines — theorems on spectra and on the structure of polynormed skew fields. This, in its turn, will lay the foundation for the central results of the general theory contained in Chapter IV.
Commutative and star Banach algebras
Authors: Helemskii A YA
Published: November 2023

shall speak of results that historically form the core of our subject and reflect that interplay of analysis and algebra which is characteristic of it. Here we shall describe, for the most part, the achievements of the “classical period” of Banach algebras (approximately 1939-1953). The results, as a rule, refer to two classes of algebras: (I) commutative and (II) those having some intrinsic symmetry the so-called star algebras. Accordingly, two closely-related theories are given (the second depends on the first). In each of them there is a climax — when there appears a construction for making “something concrete and tangible”, within the given class, from a given (abstract) algebra: in the first case this is a function algebra, in the second it is an operator algebra. The constructed algebra contains extensive information about the original algebra and, under certain additional assumptions, it is an isometrically isomorphic copy of it.
Borel’s Methods of Summability
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

Summability methods are transformations that map sequences (or functions) to sequences (or functions). A prime requirement for a "good" summability method is that it preserves convergence. Unless it is the identity transformation, it will do more: it will transform some divergent sequences to convergent sequences. An important type of theorem is called a Tauberian theorem. Here, we know that a sequence is summable. The sequence satisfies a further property that implies convergence. Borel's methods are fundamental to a whole class of sequences to function methods. The transformation gives a function that is usually analytic in a large part of the complex plane, leading to a method for analytic continuation. These methods, dated from the beginning of the 20th century, have recently found applications in some problems in theoretical physics.
Relations with the family of circle methods
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

In this chapter, we give a survey of the family of circle methods, as de- veloped by Meyer−König in [185]. These methods are related to Borel’s methods in that, under certain order conditions on the sequence {sn}, they are essentially equivalent. We shall begin with Knopp’s version of Euler’s method, as first developed by Knopp in 1922 [151], [152]. We shall then survey the Taylor methods, defined by Hardy and Littlewood in 1916 [113]. Included in the survey is Valiron’s method, given by Hardy and Littlewood in [113]. In order to assist in getting the relationships amongst these methods, Meyer−König introduced a new method in [185].
Generalizations of Borel’s methods
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

The first generalization of Borel’s method (B′) appears to have been given by Hardy [110], in 1934
Tauberian theorems − II
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

This chapter concentrates on the more delicate Tauberian theorems for Borel-type methods.
Algebraic Varieties
Authors: Yves Diers
Published: November 2023

The Zariski category RedCAlg ( k) of reduced commutative algebras over an algebraically closed field k is the most appropriate for the study of algebraic varieties over k. What are the special features of this Zariski category that make classical algebraic geometry work? It is a locally neotherian Zariski category whose initial object is simple and algebraically closed, and in which any object is reduced. We use here the notion of locally neotherian categories introduced by Gabriel and Ulmer [16) and that of algebraically closed objects introduced by Fakir [14) in any locally finitely presentable category. We prove that classical algebraic geometry can be performed on any Zariski category satisfying these properties.
Stable topological invariance of µ-constant strata
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

abstract

We begin by repeating a few definitions and theorems from earlier chapters in this book to give a coherent setting for our results.
Calculations of instability loci
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

The theory developed in Sections 9.5, 9.6, 10.2, and 10.3 shows how analysis of certain (positive) instability ideals leads to the identification of civilized submanifolds of jet space and thus, by Sections 9.2-9.4, to sufficient conditions for topological stabilityThe algorithms described in Sections 10.4-10.6 give an essentially mechanical way to compute positive instability ideals. They provide surprisingly fast computations of these ideals, given the numbers and degrees of the variables involved. None the less, all but the simplest of these computations seem to be beyond direct calculation by hand. We have used electronic computers (SUN 4, DEC 3100) furnished with computer algebra software (MAPLE) instead. This raises the question of check ability of the results obtained. In fact these results are checkable, and without too much effort. We return to the situation and notation of Section 10.4.
Necessary and sufficient conditions for stability
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

In this chapter we bring together the arguments in the previous chapters of this volume to obtain necessary and sufficient conditions for c0-stability, valid under certain conditions which we make explicit in due course. More precisely, the plan of the chapter is as follows.
The Computational Complexity of Differential and Integral Equations
Authors: Werschulz Arthur G
Published: November 2023

This book is concerned with a central question in numerical analysis: the approximate solution of differential or integral equations by algorithms using incomplete information. This situation often arises for equations of the form Lu=f, where f is some function defined on a domain and L is a differential operator. The function f may not be given exactly-we might only know its value at a finite number of points in the domain. Consequently the best that can be hoped for is to solve the equation to within a given accuracy at minimal cost or complexity. The author develops the theory of the complexity of the solutions to differential and integral equations and discusses the relationship between the worst-case setting and other (sometimes more tractable) related settings such as the average case, probabilistic, asymptotic, and randomized settings. Furthermore, he studies to what extent standard algorithms (such as finite element methods for elliptic problems) are optimal. This approach is discussed in depth in the context of two-point boundary value problems, linear elliptic partial differential equations, integral equations, ordinary differential equations, and ill-posed problems. As a result, this volume should appeal to mathematicians and numerical analysts working on the approximate solution of differential and integral equations as well as to complexity theorists addressing related questions in this area.
Dynamic Processes for Rate-Type Elastic-Visco Plastic Materials
Authors: Ionescu Ioan R , Sofonea Mircea
Published: November 2023

This chapter deals with dynamic problems for semilinear rate-type constitutive materials. The existence and uniqueness of the solution is given using arguments of linear continuous semigroups in Hilbert spaces. In the viscoelastic case an energy bound of the solution is given and it is proved that linear elasticity is a proper asymptotic theory for viscoelastic materials. An approach to perfect plasticity is also studied, by a passage to the limit. The case of rate-type materials involving internal state variables is also discussed and some monotony and fixed point methods in the study of dynamic problems are presented. Finally the evolution of the perturbations of homogeneous simple shear is analysed and the link with strain localization is given.
Elliptic Operators With Variable Coefficients
Authors: Robinson Derek W
Published: November 2023

The previous chapters were devoted to the analysis of strongly elliptic op erators associated with general representations of a Lie group. These oper ators are natural generalizations of the usual elliptic differential operators on Rd with constant coefficients and next we consider similar analogues of elliptic operators with variable coefficients. But extension of the class of operators is only possible by restriction of the representations.
A non-degenerate generalized quadrangle with lines of size four is finite
Authors: Brouwer A E
Published: November 2023

Proof. Each point is incident with the same number of lines, t say, where our aim is to show that t is finite. The hypothesis implies that there are two disjoint lines, L and M say. Consider the 16-set γ that is the union of four lines meeting both L and M.