bluuebunny/crossref_metadata_2025 huggingface dataset:
Some Preliminaries
Authors: Collins Peter J
Published: November 2023

A number of elementary results in real analysis will be used continually in the sequel. For ease of reference, we gather together here some that are more important or commonly found. Though we shall be careful, and encourage the reader to be careful, to give appropriate, sufficient conditions by means of justification, we shall not always be concerned with stating the sharpest possible results. In particular, we shall generally assume that a function is continuous when sometimes integrability over a bounded interval would suffice. Further, we shall not be concerned that differentiability is on occasion enough but normally, in these circumstances, assume that a function is continuously differentiable in the sense of the following definition. We now list some propositions, giving each a label in square brackets which we shall use subsequently for the purpose of reference. The notation (a, b) and [a, b] will be used, respectively, for open and closed intervals in ℝ.
Phase-Plane Analysis
Authors: Collins Peter J
Published: November 2023

The systems with which this chapter is concerned are those of the form satisfied by the functions x = x(t), y = y(t), where X and Y are continuous real-valued functions and dot denotes differentiation with respect to the real variable t. They are often referred to as ‘plane autonomous systems of ordinary differential equations’. Here, ‘plane’ indicates that just two functions x, y, dependent on the variable t, are involved and ‘autonomous’ that neither of the functions X, Y contains this variable explicitly. It is not in general possible to find an analytic solution to an arbitrary differential equation and, even when it is, the solution is often expressed in a form (such as a series or integral, or implicitly) which does not allow easy discussion of its more important properties. This chapter provides an introduction to a qualitative study in which one can locate properties of a differential equation without having to find a solution. Every student of differential equations should be aware of phase-plane methods. When the real plane ℝ2, with axes x and y, is used to plot solutions x = x(t), y = y(t) of (A), it will be referred to as the phase-plane. The curve traced out by (x(t), y(t)) as t varies (usually over some finite or infinite interval) is called a phase-path or trajectory. A diagram depicting the phase-paths is called a phase-diagram.
Project 9
Authors: Grandine Thomas A
Published: November 2023

A certain division of the company, namely the ACME Team Research Organization Committee for Investigating Oceans and Underwater Stuff, has sent one of its research ships, S.S. Gaussian Rule, on a voyage to gather data off the coast of Venezuela. Under orders from ATROCIOUS, Gaussian Rule has spent the last few days in the triangular region of ocean whose vertices are at 12° North latitude 65° West longitude, 12° North 64° West, and 13° N. 65° W., just north of Blanquilla Island. The ship is equipped with (among other things) two different recording devices for measuring oceanographic data. The French Remote Offshore Gauge records subtle differences in pressure at sea level, while the Temperate Offshore Analytic Device records weak shifts in the earth’s electromagnetic field.
Project 17
Authors: Grandine Thomas A
Published: November 2023

The Dependable Industrial Reprehensible Toxic Yuck company, a national chemical company, is having trouble at one of its many plants. It seems that many of the employees are developing a strange disease. DIRTY claims that the disease is caused by a virus which just happens to be going around, but the Environmental Protection Agency isn’t quite so sure, and it has hired the Corporate Liaison for Engineering Air Niceness, a division of ACME, to investigate the matter. The plant in question produces a chemical called convexitygen, a chemical with many industrial uses (it is used in the manufacture of cramerrulic, for example). The reaction which produces convexitygen takes place in a fairly simple reactor. Two compounds, arithmetic acid and boundary valuum are placed in the reactor. They react to form convexitygen. However, the convexitygen reacts with the boundary valuum to produce a waste chemical, determinantanol, which is quite toxic.
Numbers
Authors: Potter Michael D
Published: November 2023

What we have done so far has been largely introductory in nature. The aim now is to show that the axioms of basic set theory are sufficient to ensure the existence in our theory of structures with the familiar properties we expect the sets of natural, rational, and real numbers to have.
Lattices
Authors: Potter Michael D
Published: November 2023

What we now call the theory of lattices originated in the late nineteenth century from two sources: Dedekind came upon the notion of a Dualgruppe — what is now called a ‘modular lattice’ — in his investigation of the theory of divisibility; the other motivating example is the algebra of classes — theory of logical equivalence of propositions — which was studied by Boole, Schröder, and others. The work we shall be describing in this chapter is almost all the product of research in the 1930s by Birkhoff and Øre. Since then lattices have found natural application in functional analysis, projective geometry, set theory, logic, quantum mechanics, computer science, group theory, and number theory: ‘Mathematical objects have lattices as dogs, fleas’ (Isbell 1984, p. 94).
From the internal language to the topos
Authors: McLarty Colin
Published: November 2023

The reader is warned that this sequent does not say that for every generalized element yϵ B there is some xϵT A with y = fx. It says internally that every y in B is the image of some x in A, but externally this only means thatf:is epic and not necessarily split epic (see Exercise 18.5).
Factorization of polynomials
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

In order to motivate the material of this chapter, consider the fo11owing simple-looking problem: Set f(X) = 8X9 + 56X4 + 49X + 28; can f(X) be written as the product of two polynomials of lower degree with rational number coefficients? The most direct approach would seem to be to suppose that f(X) does factorize, equate it to the product of two factors with unknown coefficients, multiply the factors together and equate coefficients to get a set of equations for the unknown coefficients, solve these equations or show that they have no solution; but f(X) might be the product of polynomials of degrees 1 and 8, or 2 and 7, or 3 and 6, or 4 and 5; it is starting to look very complicated and messy.
Finite cyclic groups and finite fields
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

The main result of this chapter is that if F is a field and G is a finite subgroup of the multiplicative group of non-zero elements of F then G is cyclic. The proof is an elegant combination of elementary group theory and the theory of polynomial equations over a field. An immediate consequence of the main theorem is that if p is a prime number then the group of non-zero elements of ‘11../p”ll. under multiplication is cyclic. We shall also prove some general results about finite cyclic groups. For instance, if G is a cyclic group with n elements then G has precisely one subgroup of order d for each positive divisor d of n.
BTOP/S AS A 2-Category
Authors: Johnstone Peter T
Published: November 2023

In this chapter we shall study the remarkable 2-categorical structure of the 2-category BTop/S of bounded S-toposes, where S is a fixed topos. We shall restrict ourselves entirely to bounded geometric morphisms throughout the chapter: although some of our results can be extended to unbounded morphisms, the structure of Top/S is much more delicate, and the gain in generality does not seem worth the extra trouble that would be involved in formulating all our results so as to apply to unbounded morphisms. In this first section, our aim is to construct finite weighted limits (in the sense of 1.1.5) in BTop/S.
A First Course in Mechanics
Authors: Lunn Mary
Published: November 2023

This textbook provides a simple introduction to mechanics for students coming to the subject for the first time. The text is based on courses given to first and second year undergraduates and has been written with this audience in mind. Prerequisites are only a basic familiarity with vectors, matrices, and elementary calculus. The author's aim is to provide an understanding of Newtonian mechanics using the tools of modern algebra. The first chapters of the book introduce the fundamentals of the motion of rigid bodies: Newton's laws, forces, linear and angular momentum, and the conservation of energy. In the later chapters the theory of Lagrangian mechanics is developed and extended to cover applications to impulsive forces. Throughout, the theory is illustrated with many worked examples, and numerous exercises (some with solutions) are provided.
The classical finite rank theory
Authors: Pillay Anand
Published: November 2023

In this chapter I develop what I call the ‘classical’ part of the geometric theory, consisting essentially of results of Zilber and Cherlin-HarringtonLachlan from the 1970s and early 1980s. Early work for Buechler falls naturally into this framework.
1-based theories and groups
Authors: Pillay Anand
Published: November 2023

The notion ‘1-based’ turned up naturally in Chapter 2, as, for example, a way of describing the global behaviour of forking in a superstable theory of finite U-rank all of whose minimal types are locally modular. In this chapter we study arbitrary (stable) theories which are 1-based, and also (∞)-definable sets which are 1-based, without any assumption of finite U-rank, or even superstability. In section 1 we give various equivalences of the 1-based property. In sections 2 and 3 we show that under various additional assumptions (triviality, NDOP, few types) 1-basedness of a theory implies that the theory is ‘superstable-like’ in the sense that types have finite weight, and enough regular types exist. In section 4, we begin to study definable groups in 1-based theories, or even ∞-definable groups which are 1-based. These turn out to be essentially ‘abelian structures’, that is abelian groups equipped with predicates for subgroups. It will also be seen (in Chapter 5) that any ‘non-triviality’ of forking in a stable 1-based theory immediately gives rise to definable groups, so this analysis is quite pertinent. In section 5, we characterize the geometries attached to (locally) modular minimal groups.
Curve and Surface Fitting with Splines
Authors: Dierckx Paul
Published: November 2023

The fitting of a curve or surface through a set of observational data is a very frequent problem in different disciplines (mathematics, engineering, medicine, ...) with many interesting applications. This book describes the algorithms and mathematical fundamentals of a widely used software package for data fitting with (tensor product) splines. As such it gives a survey of possibilities and benefits but also of the problems to cope with when approximating with this popular type of function. In particular it is demonstrated in detail how the properties of B-splines can be fully exploited for improving the computational efficiency and for incorporating different boundary or shape preserving constraints. Special attention is also paid to strategies for an automatic and adaptive knot selection with intent to obtain serious data reductions. The practical use of the smoothing software is illustrated with many examples, academic as well as taken from real life.
More Smoo Hing Spline Curves
Authors: Dierckx Paul
Published: November 2023

In this chapter, we will see how the smoothing criterion of Chapter 5 can be extended in various ways.

It is often noticed that the quality of a spline approximation is somewhat inferior towards the end points of the approximation interval (see e.g. Example 5.3). This is not really surprising, considering the fact that on one side only enough information (data points) is available. Sometimes we can cope with such an unfavourable situation by imposing appropriate boundary constraints. How these can be implemented efficiently will be illustrated by a number of examples.

We will also show how the algorithms can be adapted for fitting data which describe an arbitrary curve, closed or otherwise. Finally we will make a link to computer aided geometric design (CAGD), another research area in which spline functions are extremely useful.
Fit Ing With Convexity Cons Raints
Authors: Dierckx Paul
Published: November 2023

In Chapter 6 we saw how to determine smoothing splines with B-spline coefficients that satisfy some simple equality constraints. On the other hand, certain shape preserving approximation problems result in linear inequality constraints for these coefficients. How they can be incorporated efficiently will be the topic of this chapter. : ‘.\rotably, we will describe in detail an algorithm for fitting cubic splines with local convexity constraints.

In problems of data fitting it is often required that the approximating function should mimic certain properties of the function underlying the data. If it is known. for instance, that this function is strictly positive or monotonically increasing or perhaps convex, it is desirable that the approximation has the same properties. Another reason why it can be very useful to impose monotonicity or convexity constraints is that they may prevent undesirable inflections or oscillations of the fit.

From the positiYity property (1.21) of B-splines, it immediately follows that a spline approximation s(x) in the form (1.30) is positive on the interval [a. b] if all its B-spline coefficients are non-negative, i.e. if
Sur A.Ce Itting: An Introduction
Authors: Dierckx Paul
Published: November 2023

Most of the curve fitting algorithms described have been generalized for fitting a smooth surface to a discrete set of data (xr, yr, zr). As in Chapter 3 we will first give a review of published spline fitting methods with references to available code. The software resources are basically the same (NAG. IMSL. DASL. fitpack, FITPACK, netlib/dierckx, netlib/fitpack, netlib/gcv): the yield, however, is less abundant and diverse this time. Finally, we will also briefly comment on the variational surface fitting approach as opposed to the adopted constructive approach
MORE MESH DATA SMOOTHING
Authors: Dierckx Paul
Published: November 2023

In Chapter 10 we demonstrated how least-squares and smoothing splines can be calculated very efficiently if the data are given at the nodes of a rectangular grid. In this chapter we will show how the calculations can be adapted if we want to approximate a function over a disk or a sphere, provided that the data are now given on a radius-.angle or latitude..longitude grid respectively. Furthermore we will also consider the problem of reconstructing a surface from a set of cross-sectional contours and make a link to computer aided geometric design.
Automata and Languages
Authors: Howie John M
Published: November 2023

Theoretical models of simple computing machines, known as automata, play a central role in theoretical computer science. This textbook presents an introduction to the theory of automata and to their connections with the study of languages. At the heart of the book is the notion that by considering a language as a set of words it is possible to construct automata which `recognize' words in the language. Consequently one can generate a correspondence between a hierarchy of machines and a corresponding hierarchy of grammars and languages. Professor Howie leads the reader from finite state automata through pushdown automata to Turing machines. He demonstrates clearly and elegantly the fundamental connections between automata and abstract algebra via the notions of syntactic monoid and minimal automaton. The author presupposes a basic familiarity with modern algebra, but beyond this the book is self-contained. As a result, the book will make ideal reading for students of mathematics and computer science approaching this subject for the first time.
Nilpotent Groups
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

With the general theory of the previous chapter in hand, we can begin the structural analysis of groups of finite Morley rank. The general theory occupies Chapters 6 through 9. The present chapter deals with the structure theory for nilpotent groups. For the study of simple groups on the whole, the structure of solvable groups in general is of more significance than the special case of nilpotent groups, but in any case we will see in Section 6.2 that there is a general structure theory for nilpotent groups which is remarkably close to the structure theory available in the abelian case (Exercise 9, page 72).
Permutation Groups
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

This is a long chapter on permutation groups of finite Morley rank. The reader will find numerous challenging open problems. We start with a generalization of Clifford’s Theorem to the context of groups of finite Morley rank. Our proof follows the standard proof and we do not claim any originality. We continue with an investigation of Frobenius groups where the reader can already find some number of open problems. A group B with a proper subgroup T is said to be a Frobenius group if for b ∈ B, Tb n T ≠I- 1 implies b ∈ T. As we will see, the Frobenius groups arise naturally in the theory of permutation groups.
Complex Functions as Transformations
Authors: Needham Tristan
Published: November 2023

A complex function fis a rule that assigns to a complex number zan image complex number w= f(z). In order to investigate such functions it is essential that we be able to visualize them. Several methods exist for doing this, but (until Chapter 10) we shall focus almost exclusively on the method introduced in the previous chapter. That is, we shall view zand its image was points in the complex plane, so that fbecomes a transformation of the plane.
Vector Fields and Complex Integration
Authors: Needham Tristan
Published: November 2023

We promised long ago that there was a more vivid way of understanding complex integrals than the geometric Riemann sum of Chapter 8. In this section we lay the foundations for this elegant new approach. If you are already familiar with vector calculus then you can skip this section and go directly to Section II.
Doubly indexed relational systems
Authors: Smullyan Raymond M
Published: November 2023

In this chapter we turn to some results that generalize results in recursion theory. These will be used in the next chapter to establish some generalizations of results in metamathematics. What we will now do falls under the category of generalized recursion theory, in the sense of Fitting[5].
Strong fixed point properties
Authors: Smullyan Raymond M
Published: November 2023

In this chapter we deal with the recursion property, the Myhill property, and some related “strong” fixed point properties.
A universal system and its applications
Authors: Smullyan Raymond M
Published: November 2023

We now wish to construct a “universal” system (U) in which we can, so to speak, express all propositions of the form that such and such a number is in such and such a recursively enumerable set, or more generally that such and such an n-tuple of numbers is in such and such a recursively enumerable relation. A complete knowledge of which sentences in this system are true, and which are not, would imply a complete knowledge of all formalized mathematics. But we will see that the problem is undecidable — a result that Post[13] calls “Gödel’s Theorem in Miniature.” Also, our universal system will be the basis for two fundamental results in recursion theory — the Enumeration Theorem and the Iteration Theorem, which are indispensable for many of the chapters that follow.
A menagerie of groups
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

In this chapter we introduce some of the more interesting groups that occur in applications of the theory. We restrict ourselves to a description of one of the most concrete forms of each example. In later chapters you will meet the groups again in other guises.
Actions count
Authors: Neumann Peter M , Stoy Gabrielle A , THOMPSON EDWARD C
Published: November 2023

In this chapter we discuss a famous counting theorem that is the foundation for a large area of combinatorics known as enumeration theory. Throughout the chapter n will be a finite set and G a finite group acting on 0. For g e G let

Thus fixΩ(g) consists of those elements of Ω that are in cycles of length 1 in the permutation induced by g; with the conventional rule that cycles of length 1 are not recorded, fixΩ(g) consists of those points of O that do not appear in the cycle notation for g. We write OrbΩ(G) for the set of G-orbits inn. With this notation the theorem is:
Euclidean geometry
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

It will be noted that the ratio of the (essentially positive) Euclidean distances only gives the absolute value of the invariant (δac/δab). It is this fact which gives rise to some tiresome technical difficulties in classical Euclidean geometry, which relies principally on the notion of distance. In a non- algebraic development the notion of distance has to be supported by a notion of ‘order’ or ‘between-ness’ or ‘separation’ of points on a line.
Finite groups of isometries
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

We start our discussion by proving two important lemmas about finite groups of affine transformations. These lemmas, which depend essentially on the finiteness of the group, clear a great amount of ground in preparation for the subsequent investigation.
Belief Revision
Authors: Gardenfors Peter , Rott Hans
Published: November 2023

Now suppose that, as a matter of fact, the bird caught in the trap turns out to be black. This means that you want to add the fact ->C, i.e., the negation of C, to the database. But then the database becomes inconsistent. If you want to keep the database consistent—which is normally a sound methodology—you need to revise it. This means that some of the beliefs in the original database must be retracted. You don’t want to give up all of the beliefs since this would be an unnecessary loss of valuable information. So you have to choose between retracting A, B1, B1 -> B, or A A B C. When giving up a belief you have to decide which of the reasons for this belief to retain and which to retract (this is the backward direction).
Modal Deduction with Applications in Epistemic and Temporal Logics
Authors: Cerro Luis Farinas del , Herzig Andreas
Published: November 2023

Abstract

The widespread use of non-classical logics in Artificial Intelligence makes it desirable to have automated deduction methods (and in particular resolution methods) for these logics at our disposal. In this chapter we present several automated deduction methods for modal logics. We focus on the problems of normal forming and variable skolemization, which prove to be harder than in classical logic because of the enriched language.
Euclidean geometry
Authors: Roe John
Published: November 2023

Let S be an affine space. We saw in the last chapter that if we also assume that S has the extra structure of an absolute distance |AB| between two points A and B, then it is possible to define a dot product  u • v between any two vectors u and V in S. This dot product has the four properties of linearity, symmetry, positivity, and definiteness, which are summarized in Figure 4.1.
Solid geometry
Authors: Roe John
Published: November 2023

In this chapter we will use vector methods to study the geometry of three dimensions, traditionally known as solid geometry. As we did for plane geometry, we will begin by discussing the concept of orientation. In two dimensions, an orientation specifies a ‘positive direction of rotation’ (6.1.2), and thereby divides the possible coordinate systems into right-handed and left handed. In three dimensions, we will find that an orientation makes possible a similar distinction between right-handed and left-handed coordinate systems.
Quadric surfaces
Authors: Roe John
Published: November 2023

In this chapter we will study the simplest examples of curved surfaces in three-dimensional space. Just as a curve can be defined by a locus equation f(x,y) = 0 relating two variables, so a surface can be defined by a locus equation f(x, y, z) = 0 relating three variables.
Differential geometry of curves
Authors: Roe John
Published: November 2023

Most of the geometry that we have been studying so far has been what is called algebraic geometry. Algebraic geometry studies curves and surfaces that are represented by polynomial equations — such as the equation x2 + y1 = 1 of the circle — using only algebraic facts about polynomials and their roots. For instance, we discussed the tangent lines to a conic in terms of double roots of the equation for the intersection of a line and a conic. This is a typical kind of algebraic-geometry argument.
Iterative Methods for Linear Systems
Authors: Elman Howard C
Published: November 2023

In Section 2, we introduce the basic ideas of stationary iterative methods and consider several particular examples of these, namely the Jacobi, Gauss- Seidel, SOR, and SSOR methods. We outline some results on the convergence of them, for both general matrices and those with special structure. In Section 3, we give an overview of Krylov subspace methods for systems where the coefficient matrix is symmetric.
Hierarchical Preconditioners for Elliptic Partial Differential Equations
Authors: Bank Randolph E
Published: November 2023

In this chapter we shall present an introductory survey of the hierarchical basis preconditioners for elliptic partial differential equations. In the past decade this class of methods has drawn much interest. Hierarchical basis methods, [4], [5], [19], [20], and [25], are closely related to multigrid methods [16], [17], and [22], and to the BPX class of methods, [8], [9], [23], [27], and [28). They share with them a rapid rate of convergence, although hierarchical basis methods have a logarithmic-like dependence on the size of the problem, whereas multigrid and BPX have no h-dependence and are essentially optimal in their rates of convergence, [21] and [7].
Hyperbolic Systems
Authors: Pandolfi Maurizio
Published: November 2023

Extended summary of the contribution given at the SERC Numerical Analysis Summer School, Lancaster University, July 1992.) A large variety of physical phenomena is described by hyperbolic systems. Fluid dynamics is probably the field of major relevance: compressible flows (unsteady and steady supersonic), shallow waters, hydraulic transients in ducts, etc. Also, the propagation of stresses in particular structures in space applications or through underground layers is governed by hyperbolic systems. Phenomena in astrophysics such as explosions and implosions or solar winds are described by these systems, as well as many further examples in physics and technology.
Introduction
Authors: Huxley M N
Published: November 2023

This book is written from the standpoint of analytic number theory, a subject whose name may seem contradictory. Number theory deals with the behaviour of the integers under addition and multiplication, and analysis with continuity and limit operations. The logicians have told us that any problem in mathematics can be disguised as finding integer solutions of a set of simultaneous quadratic equations. This means that we must be prepared to use any method that comes to hand. We may classify problems in number theory only by the methods used, not by the form in which they are stated. An argument is algebraic if it uses only addition and multiplication, and analytic if it uses the ordering of the integers, or the ordering of the rational numbers within the real line, which is a consequence of the ordering of the integers. From this point of view ‘elementary number theory’ is not a third category, but a description of the early levels. It is in this sense that the first part of this book is ‘elementary’.
The rational points close to a curve
Authors: Huxley M N
Published: November 2023

with denominators in a range Q ≤q <2Q. We do not require that the highest common factor of a, b and q be one, but that each rational point that can be written as (a/q,b/q) with Q ≤ q < 2Q is counted once only, however many triples of integers a, b, q give the required ratios. We can consider the triple a, b, q as a point of the projective plane, although the curve y =f(x) need not be algebraic. This is the same as counting triples (a,b,q) of integers satisfying (4.1.1) with 1 ≤ q < 2Q and highest common factor (a, b, q) = 1.
Lattice points and area
Authors: Huxley M N
Published: November 2023

The analytic method for estimating the discrepancy between the number of lattice points inside a closed curve involves dividing the curve into five or more pieces, taking local coordinates y =f(x) on each piece, and estimating the contribution ∑ p(f(m)) that each piece of the curve makes to the discrepancy. The classical estimate is Corollary 2 to Lemma 5.4.3. We use the Fourier expansion of Lemma 5.3.1, which leads to the sums considered in Chapter 8. Our first task is to use the estimates for the First Spacing Problem in Chapter 13 and the Second Spacing Problem in Chapter 16 to complete the estimation of the exponential sum, and deduce the bounds for the rounding error sum. As in the last section, we require that certain derivatives and determinants of derivatives do not vanish. We shall show that the local coordinate systems can be chosen to satisfy these conditions.
Sums with modular form coefficients
Authors: Huxley M N
Published: November 2023

Remarks The second term in (20.1.7) appears when we pass from a smoothed sum in Lemma 10.3.1 (with N≍MT  -1/3) to the sum (20.1.1). This term may be reduced by appealing to a stronger bound than Lemma 10.2.1, such as the full result of Rankin (1939) or the results of Moreno and Shahidi (1983) (Lemma 10.3.3) or Deligne (1974) (Lemma 10.3.2).
Polar Decomposition And Gauss Decomposition
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

Every real positive definite symmetric matrix x can be diagonalized, that is, there exist an orthogonal matrix u and a positive diagonal matrix d such that x = udu′. This gives a decomposition of x which we call the polar decomposition because of its similarity to polar coordinates in the plane. Another decomposition of x is the Gauss decomposition x = tt′, where t is a lower triangular matrix with positive diagonal entries. In this chapter we describe the generalizations of these decompositions to arbitrary Euclidean Jordan algebras. The main results are the formulas of Theorem VI.2.3 and Theorem Vl.3.9, which express the integral of a function on Ω in terms of the polar decomposition and the Gauss decomposition.
Tube Domains Over Convex Cones
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

In this chapter we study complex tube domains over proper convex cones. Our results are ultimately based on the simple fact that the Laplace transform of an integrable function supported in a proper cone defines a holomorphic function in the tube domain over the dual cone. After recalling some basic notions we give a complete exposition of the theory of the Bergman kernel and Bergman metric on general complex domains. Then we proceed to the explicit discussion of the Bergman and Hardy spaces of holomorphic functions on a tube domain over a proper convex cone. We also discuss the Bergman-Shilov boundary and the associated Poisson kernel for such domains.
Function Spaces On Symmetric Domains Of Tube Type
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

In the first three sections of this chapter we briefly consider certain Hilbert spaces of holomorphic functions which form a natural generalization of the Bergman space. They are of interest from the viewpoint of representation theory, since the group G(D) has natural unitary representations acting on them. We do not give a very exhaustive discussion of these results since they are special cases of results for arbitrary bounded symmetric domains. In the final section we discuss Poisson transforms of functions defined on the Shilov boundary and we prove that they satisfy the so called Hua differential equations. The results of this section are special to the tube-type case; they do not extend in a simple way to the general case.
Representations Of Jordan Algebras And Euclidean Fourier Analysis
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

Some questions in classical analysis can be considered as related to representations of Jordan algebras. We will study some of them in this chapter: Wishart distributions, Hankel transform, generalized K-Bessel function, and some zeta integrals. Let V be a simple Euclidean Jordan algebra of rank r, degree d and dimension n. Let ϕ be a self-adjoint representation of V on a Euclidean vector space E of dimension N, and Q be the associated V-valued quadratic form on E (see Section 4 of Chapter IV).
Solution techniques of partial differential equations
Authors: Rahman Matiur
Published: November 2023

We saw in the previous chapter that the motion of a fluid particle can be described by partial differential equations. In the study of potential wave theory, partial differential equations play a central role. We have seen, for example, how the velocity potential and the stream function, in fluid flow, are described by Laplace’s equation. In the following chapters the main topics will be gravity waves and tidal waves. Here again, the physics of the problems will be described by a set of partial differential equations. Thus, for the proper study of the above phenomena, it is imperative that we master the solution techniques of partial differential equations.
The Characters Of Gln Over A Finite Field
Authors: Macdonald I G
Published: November 2023

Both groups L and M are (non-canonically) isomorphic to the group of roots of unity in C of order prime to p - char. k.
Introduction
Authors: Rahman Q I , Schmeisser G
Published: November 2023

As usual, we denote by ℂ the field of complex numbers. In view of its well-known geometric interpretation, we refer to ℂ mostly as the complex planeand understand the elements of ℂ not only as numbers, but also as points.
Fundamental results on critical points
Authors: Rahman Q I , Schmeisser G
Published: November 2023

Every multiple zero of a polynomial is a critical point. How are the other critical points located relative to the set of zeros? An answer is given by the following theorem, which generalizes the above-mentioned fact about the critical points of a polynomial whose zeros are collinear (see Fig. 2.1). In spite of its simple proof, this theorem will turn out to be of central importance in the analytic theory of polynomials. It is not only a key to various results on critical points, but also has numerous applications of other kinds.
More sophisticated methods
Authors: Rahman Q I , Schmeisser G
Published: November 2023

Let be the extended complex plane. The affine mappings
Number of zeros in an interval
Authors: Rahman Q I , Schmeisser G
Published: November 2023

Let f be a polynomial of degree n with real coefficients. Expanding f about c E JR, we obtain
Derivative estimates on the unit interval
Authors: Rahman Q I , Schmeisser G
Published: November 2023

The results of this chapter are analogous to those presented in Chapter 14. Here, the information about the polynomial is given in terms of its values on certain subsets of the real line. The most important example is the unit interval. There are also some discrete sets of points which are of special interest, as we shall see.
Coefficient estimates
Authors: Rahman Q I , Schmeisser G
Published: November 2023

These inequalities have obvious interpretations for the coefficients of trigonometric polynomials. Real trigonometric polynomials and those which are non-negative deserve to be treated separately. However, in view of Proposition 12.4.5, any estimate for the coefficients of real trigonometric polynomials has a counterpart for the coefficients of self-inversive polynomials. In addition, certain estimates for the coefficients of non-negative trigonometric polynomials, when combined with Lemma 12.2.4, lead us to interesting inequalities for the coefficients of polynomials. Due to its special significance, we shall also determine the sharp upper bound for the modulus of the sum of the coefficients of a sine polynomial.
Sampling by Generalized Kernels
Authors: Stens R L
Published: November 2023

There are several possible extensions of this theorem. For example, one may use other than rectangular lattices, which in general are more efficient (see e.g. Peterson and Middleton 1962; Mersereau 1979; Mersereau and Speake 1983; Stark 1979; Higgins 1985; Butzer and Hinsen 1989; Higgins 1996, Chapter 14). However, in any case, bandlimitation is a rather restrictive condition, since a band-limited function is in particular an entire function which cannot be simultaneously duration-limited, in view of the uniqueness theorem for analytic functions.
Prerequisites
Authors: Pasini Antonio
Published: November 2023

We presume that the reader is familiar with linear algebra and with the basic concepts of group theory: cosets, Lagrange’s theorem, Sylow subgroups, inner and outer automorphisms.
The ASD Yang-Mills equation
Authors: Mason L J , Woodhouse N M J
Published: November 2023

In this chapter, we shall look at various ways of expressing the ASD condition, first as a system of first-order equations on the components of the gauge potential, and then as the commutativity condition on a Lax pair of operators. We shall then look at two second-order forms: the first involves a matrix potential that has become known as Yang’s J-matrix, although in the Euclidean and ultrahyperbolic cases, it has an older interpretation as a Hermitian metric on a holomorphic vector bundle; the second involves another second-order potential, which we call the K-matrix. The resulting partial differential equations will be the central objects of study in the rest of the book. Both Yang’s equation and the K-matrix equation are derived from Lagrangians, and therefore their solution spaces have symplectic structures. In the first section, we set the scene by looking at the corresponding equations in Maxwell’s theory.
Reduction to two dimensions
Authors: Mason L J , Woodhouse N M J
Published: November 2023

We now come to the central examples of reduction, in which the symmetry group has two-dimensional orbits in space-time and the reduced system has two independent variables. First we shall look at the two-dimensional translation groups, from which we obtain the ASD equations on a Riemann surface, the chiral equations, harmonic map equations, the KdV and NLS equations, and various parts of the nKdV hierarchies. When we add additional discrete symmetries, we get the extended Toda field equation, other cases of the harmonic condition on maps to homogeneous spaces, and the n-wave equation. Second, we shall consider other two-dimensional groups that act freely on space-time. A notable example is the reduction to the Ernst equation in general relativity, where the group is generated by a translation and a rotation. We also consider the possibility that the symmetry group has higher dimension, but does not act freely. An obvious example is the rotation group SO(3), acting on hyperplanes in lE. Here the nonsingular orbits are 2-spheres, but the group is three dimensional. One reduction by this symmetry is Liouville’s equation. Another example is the complex Euclidean group, which has two-dimensional orbits with nontrivial isotropy. Here the ASDYM equation reduces to the Toda field equation. 1
Preliminaries
Authors: Leedharn-Green C R , McKay S
Published: November 2023

This chapter contains some of the basic results and definitions that will be needed. Standard results are often quoted without proof. Sections 1.1 and 1.2 are concerned with basic group theory, and Section 1.8 with local fields.
Using Lie algebra theory to find bounds
Authors: Leedharn-Green C R , McKay S
Published: November 2023

The application of Lie algebra methods to prove results about finite p-groups and pro-p-groups has proved to be very productive. In this chapter we describe an application of these methods which gives a lower bound to the degree of commutativity of a settled group.
Uniserial p-adic space groups
Authors: Leedharn-Green C R , McKay S
Published: November 2023

Following the recipe of the previous chapter for constructing p-groups of coclass r, the first step is to construct all p-adic space groups R of coclass at most r. In order to carry this out the following problems must be solved:
The Fourier Transform And Adhm Construction
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

We have not yet met any interesting examples of solutions to the anti-self-dual (ASD) equations. In the present chapter we will remedy this and describe geometric constructions for ASD connections over two special four-manifolds: the flat four-torus and round (conformally flat) four-sphere. For connections over the four-torus we define a ‘Fourier transform’ interchanging ASD solutions on different bundles. These results are of a somewhat theoretical interest since they do not immediately yield any explicit solutions; but we will then go on to apply the same ideas to derive the ‘ADHM construction’ for connections over S4(which we regard as the conformal compactification of R’). This gives a complete and concrete description of all solutions in terms of certain matrix data, and the results will provide a prototype for the general theory of Chapter 4.
Excision And Gluing
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

This chapter brings together a number of loosely related topics from analysis, the general context being the description of solutions to differential equations, depending on a parameter, for limiting values of the parameter. We begin by considering the excision principle for the index of linear elliptic operators. As we shall explain below, this principle leads rapidly to the proof of the vital index formula given in Chapter 4 for the virtual dimension of Yang-Mills moduli spaces. We shall show that, as an alternative to the standard proof using pseudo-differential operators, one can prove this principle by introducing a suitable deformation of the differential operator. This discussion sets the scene for the more specialized geometrical topics considered in the rest of the chapter.
Perron-type, Ward-type, and convergence-factor integrals
Authors: Henstock Ralph
Published: November 2023

We prove various properties for the N-integral that are true for the Nointegral, and by induction we can assume that the S-integral has these properties.
Degenerative Parameters
Authors: Dwork Bernard
Published: November 2023

The kernel and images of π|u|χu:Wa+u→W^a are stable under the connection and so Theorem 6.9.1 provides a source of values of parameters a for which the differential eqn (5.7) becomes reducible over the field of rational functions. Such values of parameters provide opportunities for comparison with the classical literature. We shall list some examples based on the calculations of Chapter 12 and shall check Conjecture 6.3.1 in a few cases. The factor π will sometimes be disregarded in this chapter.
Formal Analytic Theory
Authors: Dwork Bernard
Published: November 2023

We first make Lemma 3.3.3 more explicit.
Non-Degeneracy of Oa
Authors: Dwork Bernard
Published: November 2023

The object of this section is to show that the specialization called for in Theorem 17.2.2(ivb) exists whenever R^=R and in each of the cases considered in Chapter 12. We do not give details for the Lauricella Fc (§ 12.4).
Existence: Recursive Methods
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Often, when referring to the direct product, we mean the generalized direct product. An important special case occurs when |W| = 3, and μ= 1. This is a tripling constructionthat forms a TS(3ν. λ)from a TS(ν λ) and an arbitrary quasigroup of order ν(or a latin square of order ν,or a TD(3, ν)).
Isomorphism And Invariants
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

In this chapter, we introduce the fundamental notion of isomorphism. WP examine the complexity of algorithms for deciding whether two triple systems are isomorphic. We then explore properties of triple systems that are invariant up to isomorphism, that can be used with varying degrees of success in distinguishing nonisomorphic triple systems.
Intersections
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

In this chapter, we examine the intersections of two triple systems on the same set of elements. We then consider the existence of disjoint Steiner triple systems that exhibit a stronger orthogonality property.
Independent Sets
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

A subset S ⊆V in a TS(v, λ) (V,B)is independentif B ⊈Sfor all BE B.An independent set is maximalif, for all x∈ V\ S, S⋃ ( x) is not independent (that is, it is maximal with respect to set inclusion). It is maximumif it has the largest cardinality of any independent set in the design. Independent sets arc often called arcs,and maximal independent sets are called complete arcs.
Nested and Derived Triple Systems
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

In this chapter, we examine extensions of triple systems to designs with larger block size. We have already seen a number of problems involving the placement of the blocks of a triple system (or partial triple system) into a ‘larger design’. For example, embedding permits us to increase the number of elements, immersing increases the index, and enclosing increases both. Here we consider operations that increase the block size.
Decomposability
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Kramer (1974) was apparently the first to consider indecomposable triple systems. For which values of νand λ does an indecomposable TS(v,, λ) exist? This problem is not completely settled.
Directed Triple Systems
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

In this and the next chapter, we examine extemions of designs (particularly triple systems), in which blocks are taken to contain orderedrather than unordered pairs. Vieprimarily consider two different types of ordered blocks. The transitive triple(a,b,c) is said to contain the ordered pairs (a,b),(a,c) and (b,c).The cyclic triple(a, b,c)contains the ordered pairs (a,b), (b,c)and (c, a). We examine directed triple systems in which the blocks are transitive triples, Mendelsohn triple systems in which the blocks are cyclic triples, and hybrid triple systems in which the blocks are permitted to be either.
Basic stock of examples
Authors: Helemskii A YA
Published: November 2023

Our collection of “on call” examples of algebras, given in Chap.I § 2.2, is no longer enough if we want to move on. In this chapter we get to know some new specific algebras and we also study more thoroughly the examples we already know. The algebras considered here, in much the same way as our subject itself, belong to neighbouring branches of analysis the theory of functions, operator theory, harmonic analysis; there are some questions in topology and in the mathematics of quantum mechanics that are also related to this same boundary layer. The application of the general theory of algebras of analysis to these disciplines is through just such algebras as these: to obtain information about a topological group it is necessary to involve its group algebra; to examine an “individual” operator, it is useful to include it in some operator algebra or other, and so on. On the other hand the material in this chapter is needed to get an informal grasp of the constructions and results of the general theory, to which we shall return in the next few chapters: there we shall have far more opportunity to show how all this works and what it turns out to be in specific situations