bluuebunny/crossref_metadata_2025 huggingface dataset:
Long-term potentiation and memory
Authors: Morris Richard G M
Published: November 2023

The discovery of long-term potentiation (LTP) transformed research on the neurobiology of learning and memory. This did not happen overnight, but the discovery of an experimentally demonstrable phenomenon reflecting activity-driven neuronal and synaptic plasticity changed discussions about what might underlie learning from speculation into something much more concrete. Equally, however, the relationship between the discovery of LTP and research on the neurobiology of learning and memory has been reciprocal; for it is also true that studies of the psychological, anatomical and neurochemical basis of memory provided a developing and critical intellectual context for the physiological discovery.
Bidirectional synaptic plasticity: from theory to reality
Authors: Bear Mark F
Published: November 2023

Theories of receptive field plasticity and information storage make specific assumptions for how synapses are modified. I give a personal account of how testing the validity of these assumptions eventually led to a detailed understanding of long-term depression and metaplasticity in hippocampal area CAI and the visual cortex. The knowledge of these molecular mechanisms now promises to reveal when and how sensory experience modifies synapses in the cerebral cortex. Keywords: long-term depression; metaplasticity; visual cortex; ocular dominance plasticity; Hebb; Bienenstock-Cooper-Munro (BCM) theory.
GluR2 protein-protein interactions and the regulation of AMPA receptors during synaptic plasticity
Authors: Duprat Fabrice , Daw Michael , Lim Wonil , Collingridge Graham , Isaac John
Published: November 2023

AMPA-type glutamate receptors mediate most fast excitatory synaptic transmissions in the mammalian brain. They are critically involved in the expression of long-term potentiation and long-term depression, forms of synaptic plasticity that are thought to underlie learning and memory. A number of synaptic proteins have been identified that interact with the intracellular C-termini of AMPA receptor subunits. Here, we review recent studies and present new experimental data on the roles of these interacting proteins in regulating the AMPA receptor function during basal synaptic transmission and plasticity.
Expression mechanisms underlying long-term potentiation: a postsynaptic view
Authors: Nicoll Roger A
Published: November 2023

This review summarizes the various experiments that have been carried out to determine if the expression of long-term potentiation (LTP), in particular N-methyl-o-aspartate (NMDA) receptor-dependent LTP, is presynaptic or postsynaptic. Evidence for a pre-synaptic expression mechanism comes primarily from experiments reporting that glutamate overflow is increased during LTP and from experiments showing that the failure rate decreases during LTP. However, other experimental approaches, such as monitoring synaptic glutamate release by recording astrocytic glutamate transporter currents, have failed to detect any change in glutamate release during LTP.
Long-term potentiation: outstanding questions and attempted synthesis
Authors: Lisman John
Published: November 2023

This article attempts an overview of the mechanism of NMDAR-dependent long-term potentiation (LTP) and its role in hippocampal networks. Efforts are made to integrate information, often in speculative ways, and to identify unresolved issues about the induction, expression and molecular storage processes. The pre/post debate about LTP expression has been particularly difficult to resolve. The following hypothesis attempts to reconcile the available physiological evidence as well as anatomical evidence that LTP increases synapse size. It is proposed that synapses are composed of a variable number of trans-synaptic modules, each having presynaptic release sites and a postsynaptic structure that can be AMPAfied by the addition of a hyperslot assembly that anchors 10-20 AMPA channels.
The Euclidean space fRn
Authors: Webster Roger
Published: November 2023

This chapter contains those results about the Euclidean space !Rn that are needed in our discussion of convexity. The reader should already be acquainted with some of these.
Convex functions
Authors: Webster Roger
Published: November 2023

Here we discuss the theory and applications of convex functions both in IR1 and IRn. Section 5.6 deals with an important convex function associated with a non-empty bounded convex set, its support function.
Intuitionistic Logic And Heyting Algebras
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

We here present a Hilbert-style formalism for the sentential calculus H due to Heyting (1930). We assume an infinite set of atomic sentences, binary connectives →, ∧, v for implication, conjunction, and disjunction respectively, and a unary connective ⌝ for negation. The axioms consist of all sentences of the following forms:
Gaggles: General Galois Logics
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

The aim of this chapter is to provide a uniform semantical approach to a variety of non-classical logics, including intuitionistic logic and modal logic, so as to recover the representation theorems of Chapters 10 and 11 as special cases.
The uniqueness of prime decompositions
Authors: Diestel Reinhard
Published: November 2023

In this chapter we explore the extent to which simplicial decompositions and simplicial tree-decompositions into primes are unique. Clearly, the extreme case of such uniqueness would be that any two prime decompositions of a graph consisted of the same set of factors, arranged in the same order. The latter, however, is almost never true: even in cases of graphs for which the factors in any two prime decompositions are the same, the order of these factors is likely to vary greatly. We shall therefore not consider this order aspect of uniqueness.
Decompositions into small factors
Authors: Diestel Reinhard
Published: November 2023

In this last chapter concerned with the theory of simplicial decompositions we shall be looking at the problem of decomposing a given graph G into induced subgraphs of cardinality strictly less than that of G, or strictly less than some given cardinal a; such subgraphs will informally be called small. (Needless to say, all our graphs will be infinite, and some will be more infinite than others.1) Most, though not all, of the decompositions we consider will be simplicial-but even when they are not, simplicial decompositions will provide the key to their construction.
Character Tables of Commutative Association Schemes
Authors: Bannai Eiichi
Published: November 2023

The purpose of this paper is to discuss many examples of commutative as sociation schemes and their character tables. Here we explain the intention of our present research direction as well as our recent results in [4,5, 7, 8,] in an informal way without referring to technical details. This paper was mostly written in June 1988. In submitting this final version in these Proceedings in June 1989, I did not try to update this paper systematically. Instead, in order to show some new development, I added several references at the end. This paper is aimed at mathematicians whose primary interest is in finite group theory or closely related areas. We should be very pleased if this paper provokes them to be interested in this subject and to join us in pursuing this research direction.A preliminary version of this paper was published in Japanese in the Proceedings of Japan Math. Soc. Algebra Symposium held at Fukui in July 1987. Another survey paper (43) on the related subjects aiming at more general readers and putting more emphasis on the role of orthogonal polynomials, is to be published in the Proceedings of NATO-AS! on Orthogonal Polynomials and their Applications held at the Ohio State University, May 22-June 3, 1989. This paper and [43] will supplement each other to some extent.
What to Solve?
Authors: Gofman Judita
Published: November 2023

Solving mathematical problmes is a favourite pastime of many people, from school students to professional research mathematicians. The aim of this book is to provide a wide variety of problems suitable for teenagers and students which will stimulate interest in mathematical ideas and methods outside the usual school syllabus. The problems vary in their difficulty, but are all intended to encourage independent investigation, to demonstrate different approaches to problem-solving, and to illustrate some of the famous problems that famous mathematicians have attempted to solve. For each problem the author has provided hints and a detailed discussion of their solution. Consequently, the book will provide a valuable teaching resource for schools, teacher training, and university mathematics courses. The problems themselves range from the easy to the unsolved and will provide much enjoyment for all those fascinated by mathematical ideas.
Integral Equations and Picard’s Method
Authors: Collins Peter J
Published: November 2023

Four main types of integral equations will appear in this book: their names occur in the table below. Suppose that f : [a, b] → ℝ and K : [a, b]2 → ℝ are continuous, and that λ, a, b are constants. We will search for continuous solutions y = y(x) to such equations. On occasion, x may range over a different domain from [a, b]; in which case, the domains of f and K will need appropriate modication. The function K = K(x, t) appearing in all four equations is called the kernel of the integral equation. Such a kernel is symmetric if K(x, t) = K(t, x), for all x, t ε [a, b]. A value of the constant λ, for which the homogeneous Fredholm equation has a solution y = y(x) which is not identically zero on [a, b], is called an eigenvalue, or characteristic value, of that equation, and such a non-zero solution y = y(x) is called an eigenfunction, or characteristic function, ‘corresponding to the eigenvalue λ’. The analogy with linear algebra is not accidental, as will be apparent in later chapters.
Existence and Uniqueness
Authors: Collins Peter J
Published: November 2023

We begin this chapter by asking the reader to review Picard’s method introduced in Chapter 1, with particular reference to its application to Volterra equations. This done, we may fairly speedily reach the essential core of the theory of ordinary differential equations: existence and uniqueness theorems. To see that this work is essential, we need go no further than consider some very simple problems. For example, the problem of finding a differentiable function y = y(x) satisfying has the solution y = (1 – x)–1 which does not exist at x = 1; in fact, it tends to infinity (‘blows up’) as x tends to 1. Further, most differential equations cannot be solved by performing a sequence of integrations, involving only ‘elementary functions’: polynomials, rational functions, trigonometric functions, exponentials and logarithms. The celebrated equation of Riccati, is a case in point, amongst the most simple examples. In Exercise 2, the reader is asked to show that the method of proof of our main theorem provides a practical method of seeking a solution of this equation. In general, the theorem provides information about existence and uniqueness without the need for any attempt at integration whatsoever.
Hilbert–Schmidt Theory
Authors: Collins Peter J
Published: November 2023

In this chapter, we shall recount some elements of the Hilbert–Schmidt theory of the homogeneous Fredholm equation where λ is a constant and the kernel K = K(x, t) is real-valued, continuous, and symmetric: We recall that an eigenvalue of (H) is a value of λ for which there is a continuous solution y(x) of (H), which is not identically zero on [a, b]. Such a y(x) is an eigenfunction corresponding to the eigenvalue λ. Notice that all eigenvalues must necessarily be non-zero. Although the material in this chapter fits nicely into our discussion of integral equations in Chapters 8–10, the pay-off comes in Chapter 12, as it is the widely applicable Sturm–Liouville equation that provides an important context where symmetric kernels naturally arise. For the next section only, we shall consider the case where eigenfunctions can be complex-valued. The reader will want to note how the theory for real-valued functions fits into this general case.
Project 2
Authors: Grandine Thomas A
Published: November 2023

Solve has just been asked to help out the Computer Operations Network For Useful Synthesis of Engineering Devices, a division of ACME which builds small electronic devices. CONFUSED has a new project currently in the mill: they are building the new ACME Widget For Useful Logarithms, a simple scientific calculator. You have been asked to help out the project team designing the AWFUL.
Project 6
Authors: Grandine Thomas A
Published: November 2023

Research teams from the Group for Organizing Treks to Logically Omitted Special Terrain are currently planning expeditions to the South Pole and the Sahara Desert. GOT LOST has decided to buy ACME tricorders to take along as research tools.
Project 14
Authors: Grandine Thomas A
Published: November 2023

The Medical And Legal Professions’ Research Associates for Computerized Technical and Industrial Complex Enzymes, a division of ACME which develops and manufactures medical products, has just come out with a new wonder drug, cramerrulic, used for treating cancer patients. It does a terrific job of controlling the growth of many kinds of malignant tumors, but it has a very undesirable side effect: even in moderate doses, it has a tendency to kill the patient. The drug has not yet been approved by the FDA, but MALPRACTICE has received permission to prescribe it to patients whose chances of survival are almost non-existent without it.
Project 16
Authors: Grandine Thomas A
Published: November 2023

ACME is currently under contract to help the Corporation Of Nuclear Electrical Devices, one of the nation’s largest commercial producers of electricity, build a laser fusion reactor. This reactor will be powered by a single strong laser pulse targeted at some small object in the middle of this device. The details of how this works are unimportant, but the characteristics of the laser pulse, at least for this particular project, are crucial. The people in CON ED’s laboratory have made several measurements of the laser pulse during the time when it is turned on, but they still have no idea what the pulse looks like as a function of time. Because of your excellent reputation in doing numerical computations, the people at CON ED have decided that they would like to have you produce a picture of this function. Your boss has agreed that you are just the person for the job.
Groups
Authors: McLarty Colin
Published: November 2023

This chapter treats groups as an example of structures in a category. It presumes some familiarity with groups, and is not a prerequisite to later chapters. We begin with a category C with all finite products, called the base category. For convenience we suppose that C has a selected product A × B, p  1, p  2 for each pair of objects A and B. If C is in fact Set the definition gives the usual groups. A group in Data is a data type together with computable multiplication and inverse rules, a group in Top is a continuous group, and in Man it is a Lie group.
Sub-objects, pullbacks, and limits
Authors: McLarty Colin
Published: November 2023

Then if there is an arrow k that makes the leftmost triangle commute, it can be composed with the top horizontal arrow to obtain h, thus making the outer square commute; while by definition of a pullback if there is an h that makes the outer square commute then there is a unique k that makes the whole diagram commute.
Functors and categories
Authors: McLarty Colin
Published: November 2023

As we begin to compare categories with one another we must always specify which category an object or arrow is in. We will speak of an object A of a category A, or an arrow f:B→   C of a category B.
The internal language
Authors: McLarty Colin
Published: November 2023

For more complex reasoning in a topos E, we introduce a notation combining variables with E arrows to make up a kind of set-theoretic language with a familiar (although not quite classical) logic. This language is called the internal language of E. The internal language, and how its terms refer to E arrows, is described in Section 14.1. The logic of this language is described in Section 14.2, and various proofs in it are given in Section 14.3.
Natural number objects
Authors: McLarty Colin
Published: November 2023

In this chapter, unless otherwise stated, we deal with a topos E that has a natural number object N, 0, s, defined below. Not every topos has a natural number object.
Topologies
Authors: McLarty Colin
Published: November 2023

In this chapter is described Lawvere and Tierney’s elementary treatment of the topologies on toposes that Grothendieck first described. We use any base topos E and, given a topology j on E we construct a topos Ej which is ‘E modified by making j-true formulas actually true’. Grothendieck toposes are described.
Sets
Authors: McLarty Colin
Published: November 2023

The distinguishing feature of sets is that ‘a set is determined by its elements’, which we express categorically by an axiom that says that every f: A → B is fully determined by its effect on global elements of A. A topos satisfying this axiom is called well-pointed, and we will refer to its objects as sets and its arrows as functions.
The effective topos
Authors: McLarty Colin
Published: November 2023

In the effective topos, Eff, described by Hyland using suggestions from Dana Scott, Church’s thesis is true in the strong sense that every function from the naturals to themselves in recursive. In fact, all functions between objects built from the natural numbers, including the rationals and the real numbers, are recursive. Thus those objects are naturally suited to modeling data types in programming languages, and Elf is a natural setting for higher order recursion theory.
An Introductory Courses in Commutative Algebra
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

This book aims to be a concise introduction to topics in commutative algebra, with an emphasis on worked examples and applications. It combines elegant algebraic theory with applications to number theory, problems in classical Greek geometry, and the theory of finite fields which has important uses in other branches of science. Topics covered include rings and Euclidean rings, the four-squares theorem, fields and field extensions, finite cyclic groups, and finite fields. The material covered in this book prepares the way for the further study of abstract algebra, but it could also form the basis of an entire course.
Sketches of an Elephant A Topos Theory Cornpendiurn
Authors: Johnstone Peter T
Published: November 2023

Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and thereby to demonstrate the overall unity of the subject. The material is organized in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text.
Finite Element Methods for Evolution Equations
Authors: Wahlbin Lars B
Published: November 2023

This chapter will treat the “standard” finite element space discretization as applied to two classes of evolutionary problems: parabolic problems and integro-differential equations. The further discretization of the time variable will also be treated. The development will be mainly descriptive and only occasionally will we dig into proofs.
Finite Element Methods for Parabolic Free Boundary Problems
Authors: Nochetto R H
Published: November 2023

In this chapter we discuss the numerical approximation of parabolic free boundary problems by finite element methods. We have chosen the classical solid-liquid phase transition as a model example but the underlying ideas apply to other free boundary problems as well. Such a transition is described by the so-called two-phase Stefan problem.
Numerical Methods for Dynamical Systems
Authors: Beyn Wolf-J iirgen
Published: November 2023

These lectures are intended to give a survey of numerical methods for analyzing dynamical systems. The growing interest in these systems, in particular in their chaotic behaviour, has stimulated an immense amount of theoretical as well as numerical investigation. Therefore we restrict ourselves to a few typical phenomena which are only first steps to chaos, but whose numerical properties are fairly well understood. Among the topics covered are the numerical computation of invariant sets such as stationary points, periodic orbits and tori, the transitions between these objects in parametrized systems and the analysis of the longtime behaviour of numerical trajectories which are generated with sufficiently small step size. In particular, we discuss in some detail the numerical computation of singular points and of homoclinic orbits via defining equations.
Mathematical preliminaries
Authors: Howie John M
Published: November 2023

Most of the mathematical equipment we shall need is quite elementary, and many readers will be able to skip lightly over this chapter. The algebraic ideas in Sections 1.3 to 1.7 are required primarily in Chapters 3 and 7, and some further bits of algebra, required only in Chapter 7, are deferred until Section 7.1.
Algebraic Computing with Reduce
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

REDUCE is one of the most widely available and simple to use computer algebra systems. It enables users to manipulate complex algebraic expressions and equations symbolically just as mathematicians and scientists do traditionally on paper, and its capabilities include differentiation and exact symbolic integration. This book provides a comprehensive introduction to REDUCE from starting to run it, through to using some of the user-contributed REDUCE packages written for specific applications. The authors' aim is to enable all students using REDUCE for the first time to gain a familiarity with the full range of REDUCE commands whilst at the same time learning something of the internal workings of REDUCE. Throughout, numerous exercises are provided to illustrate themes covered in the text as well as to encourage `hands on' working with REDUCE. Both authors are expert users of REDUCE and the text is based on their many years of teaching the system to undergraduate and graduate students. As a result, all those coming to use REDUCE for the first time will find this an invaluable tutor and guide. For more advanced users, it covers a number of aspects not included in the REDUCE manual.
Factorization and integration in Reduce
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

The purpose of this chapter is to give an introduction to the algorithms employed by two of the most important of Reduce’s algebraic packages, namely the factorizer and integrator, without, however, going in full detail into the underlying pure mathematics. We do not give details of the big float package, although it is one of the largest pieces of Reduce source code, nor of the several smaller packages such as the internal editor, matrix handler and so on.
The user-contributed Reduce packages
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

The capabilities of the basic Reduce system are being progressively extended by a library of “packages”, written mostly in symbolic mode by “users” rather than those primarily developing Reduce. These packages currently fall into two classes: those that are distributed with the system (since version 3.3), and those that are not distributed but are available to Reduce users. The former have mostly existed for longer and been more widely tested than the latter, and have a slightly more official status.
Groups of Finite Morley Rank
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

The book is devoted to the theory of groups of finite Morley rank. These groups arise in model theory and generalize the concept of algebraic groups over algebraically closed fields. The book contains almost all the known results in the subject. Trying to attract pure group theorists in the subject and to prepare the graduate student to start the research in the area, the authors adopted an algebraic and self evident point of view rather than a model theoretic one, and developed the theory from scratch. All the necessary model theoretical and group theoretical notions are explained in length. The book is full of exercises and examples and one of its chapters contains a discussion of open problems and a program for further research.
A second variety of fixed point theorems
Authors: Smullyan Raymond M
Published: November 2023

There is a result in combinatory logic known as the “second fixed point theorem,” whose statement, proof, and applications to undecidability will be dealt with in Part II of this chapter. It is based on methods of naming combinators within combinatory logic itself, and this fits in ideally with the overall purpose of this volume. Part I of this chapter is devoted to adequate naming functions. The second fixed point theorem itself has a host of variants, all of which have pleasant generalizations to sequential systems, which will be dealt with in our final chapter.
Some classical fixed point arguments compared
Authors: Smullyan Raymond M
Published: November 2023

In the last chapter we saw how fixed points are related to self-reference. We shall now look at fixed points in a larger perspective. Although no prior knowledge of mathematical logic, recursion theory, or combinatory logic is presupposed, Part I of this chapter consists of abstract versions of classical results in these three areas. We present them as problems, which the reader might have fun trying to solve. Solutions are given at the end of Part I.
Introduction to formal systems and recursion
Authors: Smullyan Raymond M
Published: November 2023

This and the next two chapters are written primarily, but not exclusively, for those readers with no prior knowledge of recursive function theory. We develop those elementary portions of the theory that will be needed for the remaining chapters. Our approach to the theory is via elementary formal systems, as defined in T.F.S. (Theory of Formal Systems). These chapters might also be of interest to readers who are familiar with recursion theory, but not with elementary formal systems, since these systems will be seen to facilitate a neat and simple proof of Gödel’s incompleteness theorem for first-order arithmetic. Also, readers who are familiar with T.F.S. might find Chapter 8 of interest in that it contains an improved method of arithmetizing metamathematics.
A survey of some group theory
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

In this chapter we survey the beginnings of the theory of groups, and we establish the language and notation that will be used throughout the book. The reader is assumed not to be entirely new to the subject and therefore some proofs will be omitted, others will merely be sketched.
A garden of G-spaces
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

This chapter is intended to exhibit a variety of group actions that illustrate the concepts introduced in Chapter 3, and to sow some seeds that will later provide food for thought. Although we limit ourselves here to little more than sketches, each of the examples is of some importance in some branch of mathematics, and will be treated in greater detail in later chapters.
Complex numbers and quaternions
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

Complex numbers first appeared in the early 17th century as a way of expressing the •impossible’ roots of an algebraic equation with real coefficients with the help of the ‘imaginary’ square root i of - 1. The representation of a complex number a+ ib by the point of the Euclidean plane with coordinates (a, b) in a rectangular cartesian coordinate system. known to us as the Argand diagram, appeared at the end of the 18th century, and was of vital importance to the development of the theory of functions of a complex variable which took place in the 19th century. Although the word ‘imaginary’ persists in the terminology of’real part’ and ‘imaginary part’, complex numbers ceased to be mysterious, and as more confidence was gained in handling them it was possible not only to use the Argand representation in reverse, as a way of assigning a complex coordinate to each point of the Euclidean plane, but also to think of extending the notion of complex numbers to ‘hyper•complex’ numbers which would serve a similar purpose in Euclidean space of higher dimension. It is these developments which concern us here.
Inversive geometry
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

Our main objective in this chapter is the further study of the space Ω = C υ{∞} under the action of the group M of Mobius transformations. which we shall write as
Time and Change for Al
Authors: Galton Antony
Published: November 2023

Outside physics and philosophy one seldom reasons much about time and change in the abstract, but on the other hand reasoning about particular times and changes—dates, processes, events, and so on—is all-pervasive both in the specialist sciences and in everyday life. No account of reasoning can properly be called complete if it does not say something about how we reason about change; and if, as in Al, the aim is to simulate or duplicate human reasoning computationally, then the account of change required must be both extensive and detailed. In this section we survey six areas pertaining to Al in which a need for temporal reasoning has been felt, introducing some of the key issues arising in these areas; in later sections we examine these issues in greater detail.
Area, Lattice Points, and Exponential Sums
Authors: Huxley M N
Published: November 2023

In analytic number theory a large number of problems can be "reduced" to problems involving the estimation of exponential sums in one or several variables. This book is a thorough treatment of the developments arising from the method developed by Bombieri and Iwaniec in 1986 for estimating the Riemann zeta function on the line s = 1/2. Huxley and his coworkers (mostly Huxley) have taken this method and vastly extended and improved it. The powerful techniques presented here go considerably beyond older methods for estimating exponential sums such as van de Corput's method. The potential for the method is far from being exhausted, and there is considerable motivation for other researchers to try to master this subject. However, anyone currently trying to learn all of this material has the formidable task of wading through numerous papers in the literature. This book simplifies that task by presenting all of the relevant literature and a good part of the background in one package. The audience for the book will be mathematics graduate students and faculties with a research interest in analytic theory; more specifically, those with an interest in exponential sum methods. The book is self-contained; any graduate student with a one semester course in analytic number theory should have a more than sufficient background.
Analytic lemmas
Authors: Huxley M N
Published: November 2023

We call the function f(x) the exponent (function), and g(x) the weight (function). A pure exponential sum or integral is one in which the weight function is unity. The first lemma enables us to insert a weight function g(x) whenever we have a uniform upper bound for a pure sum or integral. The other lemmas in this section give upper bounds for exponential integrals.
Examples of N0-categorical structures
Authors: Evans David M
Published: November 2023

This paper is an attempt to survey the known examples of countable No-categorical structures. Inevitably, this is a selective and subjective account, and few details and proofs are given. The emphasis throughout the paper is on constructions.
A Galois correspondence for countable recursively saturated models of Peano arithmetic
Authors: Kaye Richard
Published: November 2023

In this paper I will present some new results classifying the normal subgroups of the automorphism groups of countable recursively saturated models of PA. These results can be considered as a Galois correspondence between closed normal subgroups and initial segments of the model.
The Peirce Decomposition In A Jordan Algebra
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

In this chapter we introduce the Peirce decomposition first with respect to a single idempotent, then with respect to a complete system of orthogonal primitive idempotents, or Jordan frame. Much of this could be done more generally, but we restrict ourselves throughout to the case of Euclidean Jordan algebras, for which we have simple direct proofs in the spirit of our proof of the Spectral Theorem given in Chapter III. A basic result (Theorem IV.2.1) shows that one can make computations in a general Euclidean Jordan algebra in a manner rather similar to the special case of real symmetric matrices. Another important result (Corollary IV.2.7) states that, in the irreducible case, the group K acts transitively on the set of all Jordan frames. In section 3 we discuss the boundary structure of symmetric cones, this turns out to be closely related to the algebra developed in the first two sections. In the last section we give a brief discussion of the representations of a Jordan algebra; the Peirce decomposition gives rise to an important example of a representation.
Classification Of Euclidean Jordan Algebras
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

In this chapter, for the sake of completeness, we will give a proof of the classification of simple Euclidean Jordan algebras. The case of rank r = 2 was settled in Corollary IV.1.5. When r ≥ 3, such an algebra turns out to be isomorphic to a space of Hermitian matrices with entries in one of the algebras ℝ, ℂ, ℍ, or 𝕆. These algebras are characterized by the existence of a multiplicative Euclidean norm, and the key to the classification will be the Peirce decomposition and Lemma IV.2.2 stating that, for x inVij, yin Vjk, with i,j,k distinct:
Fundamental equations of motion of fluids
Authors: Rahman Matiur
Published: November 2023

The study of fluid motion and other related processes can begin when the laws governing these processes have been expressed in mathematical form. Usually, in investigating fluid flow problems, the physical situation is described by a set of differential equations. Thus the solution of the differential equations will predict the fluid flow pattern. For a comprehensive derivation of these equations, the reader should turn to standard textbooks including Aris (1962). In this book, we shall avoid rigorous mathematical development of the equations.
Hall Polynomials
Authors: Macdonald I G
Published: November 2023

Let o be a (commutative) discrete valuation ring, p its maximal ideal, k — o/p the residue field. Later we shall require k to be a finite field, but for the present this restriction is unnecessary. We shall be concerned with finite o-modules M, that is to say, modules M which possess a finite composition series, or equivalently finitely-generated o-modules M such that p’M=Q for some r>0. If k is finite, the finite o-modules are precisely those which have a finite number of elements.
Sampling Theory and Wavelets
Authors: Fischer A
Published: November 2023

Over the past decade wavelets and wavelet techniques have become increasingly important in the more applied areas of mathematics. There seems to be hardly any field untouched by the recent preoccupation with these and related concepts. Their effectiveness in data compression makes them invaluable for the compression of auditory signals and images (Mallat 1989b; Wickerhauser 1991; Benedetto and Teolis 1993), their usefulness being further enhanced by their suitability for edge detection (Froment and Mallat 1992; Antoine et al. 1993). Wavelet and more general multilevel methods have been shown to be extremely efficient for the preconditioning of linear systems stemming from Galerkin approximations for elliptic problems (Dahmen and Kunoth 1992; Jaffard 1992).
Diagrams and Orders
Authors: Pasini Antonio
Published: November 2023

According to the conventions stated in Section 1.2, we keep the symbols P and L as types for rank 2 geometries.
Diagram Graphs
Authors: Pasini Antonio
Published: November 2023

Section 3.1.2, we represent generalized digons by drawing no edges in D. Then D(Γ) is just the simple graph obtained from the labelled graph 1) by dropping all labels. For instance, in all examples of Section 3.1.3 except (5) and (6.e) the diagram graph is a string. The diagram graph of a building of type Dnlooks like its Coxeter diagram (Section 3.1.3(5)). The diagram graph of the geometry of black and white triangles of Section 2.1.2(10) is a triangle, the same as its Coxeter diagram (Section 3.1.3(6.e)).
Morphisms and Quotients
Authors: Pasini Antonio
Published: November 2023

In this section Γ and Γ ‘ are geometries over sets of types I and I’ with type functions t and t’, respectively.
Parabolic Systems
Authors: Pasini Antonio
Published: November 2023

We also write Gi,jand Gi,jfor Gi,jand Q{i,j}, for short. Needless to say that, if C = g(Q0) (g E Q) is another chamber in the same orbit of Q containing C0 and if we substitute C for C0 in the above, then the subgroups Qi, Q J etc. should be replaced by their conjugates gQig-1, gGJg-1 etc.
Reduction of the ASDYM equation
Authors: Mason L J , Woodhouse N M J
Published: November 2023

In Chapters 5-7, we shall consider the classification of the integrable systems that arise as symmetry reductions of the anti-self-dual Yang-Mills (ASDYM) equation. In this chapter, we discuss general aspects of the reduction process. 1 We describe a framework for classification of the reduced equations, and we give examples of conformal reduction in the Abelian case to illustrate how some of the basic linear equations of mathematical physics arise from the ASD Maxwell equations. This is followed by a treatment of symmetry reduction in the non-Abelian case. We investigate the invariance of connections, define the associated Higgs fields and discuss their properties. Finally we look at other ‘hidden’ symmetries of the ASDYM equation, which do not arise from point transformations of space-time.
Reductions to one dimension
Authors: Mason L J , Woodhouse N M J
Published: November 2023

Amongst the reductions to one-dimensional systems are two central families of examples. The first, the integrable motions of symmetric tops found by Euler, Lagrange, and Kovalevskaya, are historically important for their part in stimulating the development of the subject. This is particularly true of Kovalevskaya’s example, which she discovered by requiring that the solutions to the equations of motion should have no movable critical points, that is, the location of singularities other than poles should not depend on the initial conditions.
Hierarchies
Authors: Mason L J , Woodhouse N M J
Published: November 2023

So far, our investigation of integrability and self-duality has concentrated on the connection between Lax pairs and reductions of the ASDYM linear system. In this chapter, we shall look at some other fundamental features of integrable equations and consider the extent to which they reflect the geometry of the underlying ASD equations.
Analytic Theory
Authors: Dwork Bernard
Published: November 2023

In this section we consider Ω a sufficiently large extension of ℚ complete under a valuation extending the p-adic valuation of ℚ. We think of A as a (possibly variable) element of ℚm and a ∈ HΩ, an element of Ωn lying on the hyperplane H. We have B = ℤ[λ] and the field K ( ⊂Ω) must contain ℚ(π, λ, a). We will insist that K be complete. We assume that the rational prime p satisfies the condition
Value At The Origin
Authors: Dwork Bernard
Published: November 2023

The object of this section is to compute γg(0, 0) explicitly in the case of § 13.2 by means of Theorem 10.5.1.
Duality
Authors: Dwork Bernard
Published: November 2023

Let g∈R^, a ∈ H. Our object is to define a mapping with functorial properties of K˜a into W˜−a,−πs. The notation is that of § 16.2. Hypotheses (2.6) and (6.13) will not be used unless specifically indicated. The method used is that of Dwork (1964), §§ 6 and 9.
Triple Systems
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Triple systems are among the simplest combinatorial designs, and are a natural generalization of graphs. They have connections with geometry, algebra, group theory, finite fields, and cyclotomy; they have applications in coding theory, cryptography, computer science, and statistics. Triple systems provide in many cases the prototype for deep results in combinatorial design theory; this design theory is permeated by problems that were first understood in the context of triple systems and then generalized. Such a rich set of connections has made the study of triple systems an extensive, but sometimes disjointed, field of combinatorics. This book attempts to survey current knowledge on the subject, to gather together common themes, and to provide an accurate portrait of the huge variety of problems and results. Representative samples of the major styles of proof technique are included, as is a comprehensive bibliography.
Varieties and complexity
Authors: Evens Leonard
Published: November 2023

We shall use some basic facts about the prime ideal spectrum of a commutative ring. Good sources for this material are Bourbaki (1972) and Atiyah and MacDonald (1969). The latter text summarizes most of the relevant facts in exercises. In what follows, unless otherwise asserted, k is a field of characteristic p and G is a finite group.
Categories of Commutative Algebras
Authors: Diers Yves
Published: November 2023

This book studies the universal constructions and properties in categories of commutative algebras, bringing out the specific properties that make commutative algebra and algebraic geometry work. Two new universal constructions are presented and used here for the first time. The author shows that the concepts and constructions arising in commutative algebra and algebraic geometry are not bound so tightly to the absolute universe of rings, but possess a universality that is independent of them and can be interpreted in various categories of discourse. This brings new flexibility to classical commutative algebra and affords the possibility of extending the domain of validity and the application of the vast number of results obtained in classical commutative algebra.
The Geometry of Topological Stability
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

The aim of this book is to give necessary and sufficient conditions for aCoo map to be C0-stable; the aim is achieved in a wide range of dimensions via a detailed study of the geometry and topology of many classes of "generic" singularities. The methods developed for examining the topology and geometry use results from many areas of mathematics - geometric topology, stratification theory, algebraic topology, algebraic geometry, commutative algebra... - and further progress will doubtless be made from the application of deeper results from these areas. Conversely, it is to be hoped that the description of the behaviour of generic singularities will also have interesting consequences for these areas of mathematics, which are those with most interaction with singularity theory. The book describes original research; essentially none of its results has previously appeared elsewhere, either in scientific articles or in books.
Functional and Numerical Methods in Viscoplasticity
Authors: Ionescu Ioan R , Sofonea Mircea
Published: November 2023

This book presents some mathematical methods used in the study of initial and boundary problems for Bingham fluids and rate-depedent viscoplasticity. Existence and uniqueness results are presented and the behaviour of the solution with respect to the data is also studied. For some problems, the numerical approximation of the solution is also given and numerical results are presented. Elliptic variational inequalities, convex functions, monotone operators, semigroups of operators, and nonlinear evolution equations are used as mathematical instruments. The book presents a sudy of some problems of viscoplasticity using various and different mathematical methods, the mathematical results are interpreted from a mathematical point of view, and the theory is developed to deal with numerical results from possible applications in industry and technology.
The Local Fermat Problem
Authors: Ribenboim Paulo
Published: November 2023

Our aim is to investigate the solution of Fermat’s equation in the local fields Qp of p-adic numbers. In order to have a mostly self-contained presentation, we shall begin with preliminaries about the resultant and Hensel’s lemma. First, we recall some notation and well-known facts.
Connections, Definite Forms, and Four-manifolds
Authors: Petrie Ted , Randall John
Published: November 2023

The central theme of this book is the study of self-dual connections on four-manifolds. The authors have adopted a topologists' perspective and so have included many of the explicit calculations using the Atiyah-Singer index theorem as well as presenting arguments couched in terms of equivariant topology. The authors' aim is to present moduli space techniques applied to four-manifolds and to study vector bundles over four-manifolds whose structure group is SO(3). Results covered include Donaldson's proof that the only positive definite forms occur as intersection forms and the results of Fintushel and Stern which show that the integral homology cobordism group of integral homology three-spheres has elements of infinite order. Little previous knowledge of differential geometry is assumed and so postgraduate students and research workers will find this both an accessible and complete introduction to an area that is currently one of the most active in mathematical research.
Algebraic Computing in General Relativity
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

Based on lectures given at a summer school on computer algebra, the book provides a didactic description of the facilities available in three computor algebra systems - MAPLE, REDUCE and SHEEP - for performing calculations in the algebra-intensive field of general relativity. With MAPLE and REDUCE, two widespread great-purpose systems, the reader is shown how to use currently available packages to perform calculations with respect to tetrads, co-ordinate systems, and Poincare` gauge theory. The section on SHEEP and Stensor, being the first published book on these systems, explains how to use these systems to tackle a wide range of calculations with respect to tackle a wide range of calculations in general relativity, including the manipulation of indicial formulae. For the researcher in general relativity, the book therefore promises a wide overview of the facilities available in computer algebra to lessen the burden of the lengthy, error-prone calculations involved in their research.
The Additive Divisor Problem and Exponential Sums
Authors: Jutila M
Published: November 2023

involving the divisor function d(n) and some functions f and g satisfying certain regularity conditions. Because it is easier to work with smoothed sums, we equip the outer sum in Weyl’s method - taken over squares of short sums - with a suitable weight function η ( n), which .is a smoothed version of the characteristic function of the interval of summation.
Representing integers as sums of squares and Hypothesis H
Authors: Pihko Jukka
Published: November 2023

Lagrange’s Theorem states that every positive integer can be written as the sum of four squares of integers. However it is known that we don’t need to use all possible squares of integers, and that even some infinite sets can be missed out.