bluuebunny/crossref_metadata_2025 huggingface dataset:
Approaches to problem solving
Authors: Gofman Judita
Published: November 2023

‘Devising a plan, conceiving the idea of an appropriate action, is the main achievement in the solution of a problem’, claims Polya in his famous book on problem solving, How to Solve It. According to Polya, a good idea is a piece of good fortune and we have to deserve it by perseverance: An oak is not felled at one stroke. If at first you don’t succeed, try, try again. It is not enough to try repeatedly. We must try different means, vary our trials.
Differential and Integral Equations
Authors: Collins Peter J
Published: November 2023

Differential and integral equations involve important mathematical techniques, and as such will be encountered by mathematicians, and physical and social scientists, in their undergraduate courses. This text provides a clear, comprehensive guide to first- and second-order ordinary and partial differential equations, whilst introducing important and useful basic material on integral equations. Readers will encounter detailed discussion of the wave, heat and Laplace equations, of Green's functions and their application to the Sturm-Liouville equation, and how to use series solutions, transform methods and phase-plane analysis. The calculus of variations will take them further into the world of applied analysis. Providing a wealth of techniques, but yet satisfying the needs of the pure mathematician, and with numerous carefully worked examples and exercises, the text is ideal for any undergraduate with basic calculus to gain a thorough grounding in 'analysis for applications'.
Project 1
Authors: Grandine Thomas A
Published: November 2023

The BUG is a fully functioning device, at least in terms of what it was designed to do. Nevertheless, your predecessor was unable to explain the behavior of this routine (hence the reason for his dismissal). Your boss would really like to know just why this routine for the BUG works the way it does on 32 bit machines. Check it out on your computer and explain the results.
Rudimentary structures in a category
Authors: McLarty Colin
Published: November 2023

In this chapter we study axioms for a category, and possible structures within it. We introduce the ‘generalized element’ notation, which allows us to restate some of our axioms and definitions in a more familiar form.
Products, equalizers, and their duals
Authors: McLarty Colin
Published: November 2023

Definitions more intricate than that of a monic or a terminator are easier to give using diagrams and commutativity. We will not be formal about these. A diagram is an arrangement of objects and arrows, such as: We say that the objects occupy vertices of the diagram, and the arrows occupy its edges.
Natural transformations
Authors: McLarty Colin
Published: November 2023

Given a parallel pair of functors F, G: A→B, a natural transformation from F to G is a family of arrows vA: FA→ GA, one for each object A of A, such that for every arrow f: A→ A' of A the following square commutes: Suppose that A is Cartesian closed and take any object A of A.
Slice categories
Authors: McLarty Colin
Published: November 2023

When dealing with sets we often work with sets of sets. In geometry we often deal with spaces that vary over another space (as with fibre bundles or sheaves). And high-level programming languages have dependent types, data types which themselves depend on parameters in other data types. All of these are represented categorically by indexed families of objects.
Basics
Authors: McLarty Colin
Published: November 2023

A topos is a Cartesian closed category with a sub-object classifier. A subobject classifier is an object Q and a global element t: 1 → Q such that for any monic s:S →A there is a unique arrow xS:A → that makes the following a pullback: Throughout the rest of this chapter we assume that we are working with a topos E.
The fundamental theorem
Authors: McLarty Colin
Published: November 2023

The fundamental theorem of toposes says that every slice of a topos is also a topos and, for every arrow f: A→ B of E, the functorf:*:E/B→ E/A has a right adjoint 77f and a left adjoint Xf. This makes indexed families of objects and arrows in a topos easy to handle, and it proves a great deal about the structure of a topos. The last two sections of this chapter discuss aspects of Boolean logic in toposes, and show that in toposes the axiom of choice implies the law of excluded middle. Unless otherwise noted, objects and arrows are understood to be those of a fixed topos E.
Relations in regular categories
Authors: McLarty Colin
Published: November 2023

Regular categories, defined below, support a nice theory of relations. We follow Freyd and Scedrov (1990) in extending any regular category C to a regular category Map(C) with quotients for all equivalence relations. Then we give a necessary and sufficient condition for Map(C) to be a topos. We begin by assuming that Chas allfinite limits, and that it is regular, once we have defined that.
Lagrangian mechanics
Authors: Lunn Mary
Published: November 2023

In previous chapters we have seen how the vector equations of motion are formulated both for particles and for rigid bodies. There are a limited number of examples in which a full analytic solution is possible. At some stage in the development of each solution it is necessary to think about a set of scalar coordinates which are appropriate to the geometry of the system under consideration. Sometimes this choice of coordinate system is clear cut. There is no need to use anything other than Cartesian corrdinates in the case of a projectile. On the other hand an angular coordinate is obviously a more suitable choice for a particle moving on a circular path. Consideration of the motion of a planet about a star is neatly tackled using plane polar coordinates given that such motion occurs in a plane. We should ask whether it is possible to cut out the initial stage of setting up the equations of motion using vectors and to move straight to the equations involving a chosen coordinate system. Lagrangian mechanics provides the means to do precisely that.
The Theory and Numerics of Differential-Algebraic Equations
Authors: Rheinholdt Werner C
Published: November 2023

These notes introduce some typical applications and basic properties of differential-algebraic systems of equations (DAEs) and then present an overview of recent, new existence theories for such systems based on differential geometric considerations and on a numerical approach derived from these theories. In the presentation the stress is on general concepts, results and applications rather than on detailed proofs.
Bivariate Splines
Authors: Dierckx Paul
Published: November 2023

The theory of univariate splines has been extended in different ways to functions of more than one variable. Among all these generalizations, the so-called tensor product splines (for an introduction see e.g. de Boor (1978) or Cox (1975)) are certainly the ea.5iest to handle. We will describe how the algorithms of Chapter 1 can be used in a very straightforward way for efficiently calculating with these functions. The resulting evaluation and differentiation algorithms are the basis of all tensor product spline packages. Further we will also consider a less trivial application (for use in Chapter 11), i.e. the computation of the spherical harmonic coefficients of a tensor product spline. Finally, we will briefly comment on the graphical representation of bivariate splines and illustrate how knot insertion might be useful in that respect.

Tensor product splines also have disadvantages as will be apparent in further chapters of this book. Additionally we will therefore briefly discuss and compare two other types of spline generalizations which have been used for surface fitting purposes. They share the property that they can both be defined on arbitrary triangulations. For this reason they are definitely more flexible than tensor product splines which are tied to rectangular domains. As we will see, however, they are also far more complex and therefore computationally less attractive.
Visual Complex Analysis
Authors: Needham Tristan
Published: November 2023

This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.
The structure of totally categorical structures
Authors: Hodges Wilfrid
Published: November 2023

Among the many programmes that go to make up research in model theory today, one of the most elegant is the programme to catalogue all the countable models of totally categorical theories. Morley (1965, 1967) and Baldwin and Lachlan (1971) laid the foundations. Between about 1976 and 1984, Zil’ber made enormous progress, but a large part of his work was unavailable in English until the recent translation of his doctoral thesis (Zil’ber, 1993). In the meantime, hints were published (Zil’ber, 1979, 1984b, and elsewhere), enough for workers in the West to carry the work forward. Hrushovski, Ahlbrandt and Ziegler all published papers that cast fresh light. More is on its way into print, and indeed I know that there are people who know many things not yet known to me. So this account makes no claim to completeness. I aim only to tell a good story.
Permutations and the axiom of choice
Authors: Truss J K
Published: November 2023

The purpose of this paper is to explore the connections between permutation groups, definability, and the axiom of choice (AC). The fact that these three topics are closely related is perhaps rather well known, but it is possible to bring out into the open some of the precise connections, which may be spelt out in quite a systematic way. To start from a particular point of view, in the original models for the negation of the axiom of choice there was an explicit use of permutation groups. I refer to the Fraenkel-Mostowski method. Moreover, this was developed to a high degree of elegance, particularly in the work of Mostowski where he was able, by careful choice of the group and the ‘support structure’, to illustrate non-implications between various weak versions of AC. For instance, he found a model in which every set can be linearly ordered (the ordering principle) but AC fails (Mostowski, 1939), and (1945) he investigated the connections between versions of AC for families of finite sets. More sophisticated constructions were used by other authors, for instance Lauchli (1964), who gave a model in which any family of non-empty finite sets has a choice function, but the ordering principle fails.
Relational structures and dimensions
Authors: Wagner Frank 0.
Published: November 2023

It had been conjectured that, in every countable stable theory which is categorical in some power, one can define finitely many classical geometries (cf. Example 6.2) which determine the structure. More precisely, there were two conjectures.
Theory of surface waves
Authors: Rahman Matiur
Published: November 2023

As we know in practice, real water waves propagate in viscous oceans over an irregular rough bottom of varying permeability. However, a striking feature of the water waves in oceans is that in most cases the main body of the fluid motion is nearly irrotational such that the viscosity which causes the rotation may be negligible. Another remarkable fact is that water can be reasonably regarded as incompressible and, consequently, a velocity potential and a stream function should exist for wave propagation in oceans. We have already remarked in Chapter 2 about these two mathematical entities and their importance in the solution process of water wave mechanics.
Inverse scattering transforms and the theory of solitons
Authors: Rahman Matiur
Published: November 2023

The theory of solitons and the method of inverse scattering transforms have received extreme interest from applied mathematicians and physicists in recent years, due to the most recent and fascinating achievements in the domain of applied mathematics. The KdV equation which leads to the theory of solitons has been the object of intensive scientific research in different branches of physics. We have already seen in Chapter 9, that under certain periodic boundary conditions, the KdV equation admits solitary wave solutions. These solitary waves which travel at different speeds may pass through one another without any change in the original structures; the only effect of the interaction is a change of phase.
Wavelets
Authors: Holschneider M
Published: November 2023

Wavelets analysis--a new and rapidly growing field of research--has been applied to a wide range of endeavors, from signal data analysis (geoprospection, speech recognition, and singularity detection) to data compression (image and voice-signals) to pure mathematics. Written in an accessible, user-friendly style, Wavelets: An Analysis Tool offers a self-contained, example-packed introduction to the subject. Taking into account the continuous transform as well as its discretized version (the ortho-normal basis) the book begins by introducing the continuous wavelets transform in one dimension. It goes on to provide detailed discussions of wavelet analysis of regular functions, tempered distributions, square integrable functions, and the continuous wavelet transform. Throughout, the language of group theory is used to unify various approaches. Profusely illustrated and containing information not available elsewhere, this book is ideal for advanced students and researchers in mathematics, physics, and signal processing engineering.
Multi-Resolution Analysis
Authors: Holschneider M
Published: November 2023

In this chapter we present the general framework for the construction of an orthonormal basis of wavelets over the real line ℝ, the discretized real line ℤ, the circle 𝕋, and the discretized torus ℤ/Nℤ.
Fractal Analysis And Wavelet Transforms
Authors: Holschneider M
Published: November 2023

This chapter is devoted to a short introduction to fractals in general and their analysis through wavelet transforms. We may distinguish at least three aspects in fractals an underlying dynamical system, global self-similarity, local self-similarity.
Analytic Theory of Polynomials
Authors: Rahman Q I , Schmeisser G
Published: November 2023

This text presents easy to understand proofs of some of the most difficult results about polynomials. It encompasses a self-contained account of the properties of polynomials as anlytic functions of a special kind. The zeros of compositions of polynomials are also investigated along with their growth and some of these considerations lead to the study of analogous questions for trigonometric polynomials and certain transcendental entire functions. The strength of methods are fully explained and demonstrated by means of applications.
Derivative Sampling-A Paradigm Example of Multichannel Methods
Authors: Higgins J R
Published: November 2023

This chapter is a short account of derivative sampling, the name usually given to the process of reconstructing a suitable function from a knowledge of its values and values of some of its derivatives at a discrete subset of its domain. Here, particularly in §3.4, we shall also take it to mean the dual problem of constructing a function which is to have given values and given derivative values on a given set (more remarks about this kind of duality can be found in Higgins 1996, p. 2, et seq.).
Approximation by Translates of a Radial Function
Authors: Dyn M
Published: November 2023

The study of approximation schemes based on translates of a radial function to a set of points (centres) in ℝ d is a central subject in multivariate approximation theory. Here we study the L ∞-approximation orders of such schemes, first for centres constituting a regular grid, and then for quasi-uniformly scattered centres.
Recovering Geometries from Diagrams
Authors: Pasini Antonio
Published: November 2023

We now start the job we promised to do at the end of Chapter 3: given a diagram, we try to describe the geometries belonging to that diagram, either classifying them, or at least characterizing them by other axioms of ‘global’ nature which are presumably more familiar to the reader. The tools we can use at this stage are very few and seemingly weak: we only have some results on disconnected diagrams (in particular, the direct sum theorem), the concept of geometric poset for string diagrams, the construction of Grassmann geometries and a number of results on intersection properties. However, we can already obtain many characterizations, in spite of the poverty of our equipment.
The General Theory of Integration
Authors: Henstock Ralph
Published: November 2023

Written by one of the subject's foremost experts, this is the first book on division space integration theory. It is intended to present a unified account of many classes of integrals including the Lebesgue-Bochner, Denjoy-Perron gauge, Denjoy-Hincin, Cesaro-Perron, and Marcinkiewicz-Zygmund integrals. Professor Henstock develops here the general axiomatic theory of Riemann-type integration from first principles in such a way that familiar classes of integrals (such as Lebesgue and Wiener integrals) are subsumed into the general theory in a systematic fashion. In particular, the theory seeks to place Feynman integration on a secure analytical footing. By adopting an axiomatic approach, proofs are, in general, simpler and more transparent than have previously appeared. The author also shows how one proof can prove corresponding results for a wide variety of integrals. As a result, this book will be the central reference work in this subject for many years to come.
Generalized hypergeometric functions
Authors: Dwork Bernard
Published: November 2023

Hypergeometric functions have occupied a significant position in mathematics for over two centuries. This monograph, by one of the foremost experts, is concerned with the Boyarsky principle which expresses the analytical properties of a certain proto-gamma function. Professor Dwork develops here a theory which is broad enough to encompass several of the most important hypergeometric functions in the literature and their cohomology. A central theme is the development of the Laplace transform in this context and its application to spaces of functions associated with hypergeometric functions. Consequently, this book represents a significant further development of the theory and demonstrates how the Boyarsky principle may be given a cohomological interpretation. The author includes an exposition of the relationship between this theory and Gauss sums and generalized Jacobi sums, and explores the theory of duality which throws new light on the theory of exponential sums and confluent hypergeometric functions.
Foundations
Authors: Helemskii A YA
Published: November 2023

Strictly speaking, we shall only start on the subject of this book after we have introduced an additional structure – multiplication — into the objects of linear functional analysis — Banach spaces and some more general spaces; this is precisely what we shall talk about in the first definition of the first chapter. But first, in Chapter 0, it is necessary to indicate the stock of background material which is important: for us as a foundation.

Some of this material is widely known and is contained in standard textbooks but there is some which can be found only in specialized monographs and articles in journals. Therefore the amount of detail given in different: places in this chapter is very varied.
The polynomial Waring and Goldbach problems
Authors: Effinger Gove W , Hayes David R
Published: November 2023

From 1930 to 1932, Leonard Carlitz held a National Research Fellowship. He spent a substantial portion of his fellowship at Cambridge University, where he met Raymond Paley, a brilliant young English mathematician of about the same age as himself. In the decade just gone by, Hardy and Littlewood had captured the interest of the mathematical world with a remarkable series of papers on two classic problems in additive number theory—the Waring problem and the Goldbach problem. The new technique that they relentlessly focused on these problems was the circle method, first introduced by Hardy and Ramanujan in 1918 to derive an asymptotic expansion for the partition function.
The polynomial Waring problem
Authors: Effinger Gove W , Hayes David R
Published: November 2023

Our job now is to analyse the integral in (8.5). Following the circle method approach to the classical Waring problem, which involves the Farey dissection of the unit interval, we decompose our fundamental domain D as follows:
Flatness Proper Ties
Authors: Yves Diers
Published: November 2023

Flatness properties are among the basic properties of Zariski categories. Up to now we have used the concept of flat morphism, but the first concept is that of flat object. An object A in A is flat if the endofunctor A ∐ (−) of A preserves monomorphisms. Then a morphism f: A→ B in A is flat if (B,f) is a flat object in A/ A, and an object A in A is absolutely flat if any morphism f: A→ B is flat in A. In an arbitrary Zariski category simple objects need not be absolutely flat, and therefore the two notions of von Neumann regular objects and absolutely flat objects are generally different. Some Zariski categories can have special flatness properties such as the following. Any object is flat, as in the category CAlg(k) of commutative algebras over a field k. Any object is absolutely flat, as in the category RegCRng of von Neumann regular commutative rings. Any simple object is flat, as in the category CRng of commutative rings. Any local flat epimorphism is an isomorphism, as in the category CAlg (A) of commutative algebra over a ring A. In an arbitrary Zariski category, a flat object A is faithfully (reflexively, properly) flat if the endofunctor A ∐ (−) reflects monomorphisms (isomorphisms, the terminal object). These three notions are generally different, but in some Zariski categories, such as CAlg (A), they coincide.
Proper maps and function space topologies
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

In this chapter we first collect together a number of topological facts concerning proper maps and function space topologies. Indeed, proper maps will appear frequently in this book, both as representatives of germs and in connection with global questions. Most of the results are taken from the literature, but the references are somewhat scattered and do not include all we need, so we give a unified treatment. We treat Sections 3.2-3.4 as reference sections for results of this kind needed in the rest of the volume.
Disruptive germ classes
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

In this chapter we present our key ideas for relating questions of c0-stability to more concrete problems about the topology of perturbations of maps. Our approach uses some ideas from the earlier work of May (1973) but there are significant improvements.
The Flow of the Bingham Fluid with Friction
Authors: Ionescu Ioan R , Sofonea Mircea
Published: November 2023

This chapter deals with the study of some boundary-value problems describing the flow of the Bingham fluid with friction through a bounded domain Ω in ℝ. Existence and uniqueness results are given using elliptic variational inequality arguments. The blocking phenomenon is also studied and the link is given between the yield limit, the friction coefficient and the external forces for which the flow of the Bingham fluid is blocked. An iterative finite-element method for the approximation of the solution is presented and applied to wire-drawing processes.
Flocks of cones and generalized hexagons
Authors: Bader L
Published: November 2023

The aim of this paper is to discuss the possibility of defining generalized hexagons from flocks of cones by reversing the previous construction of Kantor.
Group partitions, affine spaces and translation groups
Authors: Batten L M , Brown J M N
Published: November 2023

Affine and projective geometries developed ‘classically’ as the study of invariants of certain groups of projective collineations. The equivalence of desarguesian projective and affine geometries to the well-known constructions from vector spaces over skew-fields was one of the first major results by way of a characterization. For more information on this, the reader is referred to Hilbert (1899), Veblen and Young (1910) and Dembowski (1968).
An embedding theorem for partial linear spaces
Authors: Blokhuis A , Calderbank A R , Metsch K , Seress A
Published: November 2023

We obtain the corollary that A(26, 8, 5) = 30, since Schellenberg (1975) has used a computer to prove that (B(5, 1) does not exist. We sketch a coding theoretic proof of this result. The partial linear space B(5, 1) is a symmetric group divisible design.
Generalized Reed-Solomon codes and normal rational curves: an improvement of results by Seroussi and Roth
Authors: Storme L , State J A Thas
Published: November 2023

In particular, we will consider k-arcs which are subsets of a normal rational curve, that is, the k-arcs which correspond to generalized Reed-Solomon codes and generalized doubly-extended Reed-Solomon codes (Seroussi and Roth 1986).
Coupling Constants for p-Adic Groups
Authors: Plymen R J
Published: November 2023

The work of Vaughan Jones has brought into prominence the theory of II1 factors, their traces and coupling constants. Factors of type II1 were discovered by Murray and von Neumann. They are weakly closed, star algebras of operators on a Hilbert space, with centre consisting of the scalars, and equipped with a finite, positive, trace functional. Such a factor N provides a scale by which to measure the relative dimension of the Hilbert space H on which it acts. We write this number as dimN(H). In this case there are no restrictions on the number. Indeed it may vary continuously from 0 to ∞. Jones varied this idea and looked at a pair of factors N ⊆ M considering the dimension of M relative to its subfactor N. This he defined as [M : N] = dim N (H)/ dimM(H), showing it to be independent of the Hilbert space. The importance of this idea became evident in the restrictions found on the index.
Iwasawa Theory, Factorizability and the Galois Module Structure of Units
Authors: Solomon David R
Published: November 2023

The aim of this paper is to present a new, p-adic approach to a certain problem concerning the structure of algebraic number fields. The problem is that of describing the group of units (more generally the S-units) of an extension field as a module for the Galois group. Our approach involves the application of Iwasawa's theory of Zp-extensions. The main module-theoretic tool will be the concept of factor equivalence. This is an equivalence relation between lattices acted upon by a finite abelian group G and is a weakening of (local) isomorphism as G-modules. The idea was first introduced in [19]. Much of the fundamental work is due to Fröhlich and throughout its development the emphasis has been on applications to questions of Galois module structure (see [19] and also, for example, [3]). Factor equivalence was first applied to the study of units in [10]. This paper, together with (9], are the main references for Sections 1 and 3 in which the basic ideas and results of factorizability and factor equivalence are explained. They should be consulted for further details and proofs of some results. (Our notation will differ somewhat from theirs however.
p-Adic Fourier Series
Authors: Woodcock C F
Published: November 2023

Throughout Z p and Q p will respectively denote the ring of p-adic integers and the field of p-adic numbers (for p prime). Thus Z p is compact and is the closure of Q p (indeed of N) in Q p . We denote by C P the completion of the algebraic closure of Q P with respect to the p-adic metric. Let V p denote the p-adic valuation of C p normalized so that V p (p) = 1.
Analytic functional calculus via integral representations
Authors: Eschmeier Jörg , Putinar Mihai
Published: November 2023

In a unital, commutative Banach algebra A there is a very natural notion of a joint spectrum for systems of elements in A. Let a = (a  1, …, a  n) ∑ A  n. Then the joint spectrum σ(a) of a is defined as the set of all n-tuples z ϵ ℂn for which the ideal generated by z  1 − a  1, …, z  n − a  n in A is proper, or equivalently, as the set σ (a) = â(ΔA), where In the fifties, Shilov (1953), Waelbroeck (1954), and Arens and Calderon (1955) showed how to construct an analytic functional calculus for finite systems of elements in A.
Spectral analysis on Bergman spaces
Authors: Eschmeier Jörg
Published: November 2023

Among the examples on which the techniques developed throughout our book can be tested, there is a privileged one: the class of tuples of analytic Toeplitz operators acting on Bergman spaces over bounded pseudoconvex domains in ℂn. This is for two reasons: first because of the maturity of the function theory on Bergman spaces, and second because of the recent advances in the spectral theory of single Toeplitz operators on one-variable Bergman spaces. To be more specific, the fundamental L  2-estimates for the a-operator proved by Hormander (1965) permit the solution of basic division and interpolation problems in spaces of analytic functions with control on the L  2-growth at the boundary (see for instance Mityagin and Henkin 1971; Skoda 1972; Ohsawa 1988). On the other hand, the spectral picture of Toeplitz operators acting on one-dimensional Bergman spaces is well developed (see Axler 1988).
Transcendental Numbers and Functions of Several Variables
Authors: Waldschmidt Michel
Published: November 2023

We start by two density problems: one was raised by Sansuc and deals with algebraic number fields, the other is a question of Mazur and deals with abelian varieties. While the first has just been completely solved by D.Roy, only partial answers have been obtained so far for the second. In both cases the main tool is the theorem of the algebraic subgroup. We show tha.t this theorem enables one to recover Baker’s result on linear combinations of logarithms of algebraic numbers in at least four different ways. Another consequence, once more due to R. Roy, concerns matrices whose entries are linear combinations of logarithms of algebraic numbers.
Estimates for Discriminants and Resultants of Binary Forms
Authors: Evertse Jan-Hendrik
Published: November 2023

The height H(P) of a polynomial ρ with complex coefficients is defined as the maximum of the absolute values of these coefficients.
K-nombres de Pisot et de Salem
Authors: Bertin M J
Published: November 2023

A. M. Bergé et J. Martinet [1] ont introduit en 1987, apropos de régulateurs relatifs, la notion de hauteur relative d’un nombre algebrique par rapport a un corps de nombres K.
Response
Authors: Ribenboim Paulo
Published: November 2023

I am very happy with all my friends who are here, and who have come to Kingston to hear many outstanding speakers, and perhaps also a little bit to comfort me as I approach my retirement. So I will begin with a warning: I am not yet retiring! In fact, I will only do so two years from now. I hope not everyone here is saddened by that news! Looking back to all my years of mathematical activity, I see little merit in what I have done. I will explain why.
General principles of MPT
Authors: Grenander Ulf
Published: November 2023

The regular structures introduced in Part I, configuration spaces, image algebras, patterns, are qualitative in nature. For example, a configuration is either regular or not. In Chapters 7-11 we shall relax the regularities, allowing them to be gradual as measured by their probability of appearing in the ensemble we are observing. Expressed otherwise, we shall measure the degree of regularity or irregularity, see the morphogenetic device in Section 4.3.9.3. This metric approach, metric pattern theory, or MPT, leads us to introduce probability measures on spaces that contain and may be larger than the regular structures; see Note 7.A.
Deformation mechanisms
Authors: Grenander Ulf
Published: November 2023

If all images in a pattern were presented in their pure form to the observer the task of pattern analysis would be drastically simplified compared to what is usually the case: the observed images are degraded versions of the pure ones. It would be unrealistic to sidestep this issue and we must therefore study pattern deformations in some detail. Let us start with some generalities.
Applications of Hall sets
Authors: Reutenauer Christophe
Published: November 2023

We begin by introducing Lyndon words and the Lyndon basis of the free Lie algebra; this is a particular Hall set, with special properties, such as triangularity of the change of basis. The dual basis of the Poincare-BirkhoffWitt basis associated to a Hall basis may be computed recursively, by using the shuffle product; this is done in Section 5.2. In the next section a special Hali basis is constructed, which is compatible with the derived series of the free Lie algebra. The special properties of Lyndon words, with respect to the alphabetical order, are also true for Hall words once the appropriate order on words is defined: to obtain it, one factorizes each word, and then orders sequences of Hall words alphabetically (Section 5.4). In the final section, we show how Hali sets lead to the construction of variable-length codes, with various synchronization properties.
Open induction, Tennenbaum phenomena, and complexity theory
Authors: Kaye Richard
Published: November 2023

This paper collects together several ideas concerning weak systems of arithmetic related to the theory !Open of Shepherdson (1964, 1965) and the theory I E  1 of Wilmers (1985), and describes some of the author’s investigations in this area over the last few years. Its two main themes are encapsulated by the following questions: in what way do these theories relate to the computational complexity of natural problems? And: how might one go about proving independence results in this realm? To set the scene and to discuss these questions in more detail we must first give some definitions.
Integer Parts of Real Closed Exponential Fields
Authors: Ressayrc J -P
Published: November 2023

This paper is an extended abstract: the detailed proofs an being prepared for future publication. We define real exponential fields as ordered fields with a function 2x satisfying a set E of four basic axioms. We show that every real closed exponential field in this sense possesses an “integer part”, that is a discrete subring Z such that z+ is closed under 2x, and every element x of the field has an “integer part” ⌊ x ⌋ ∈ Z, such that ⌊ x ⌋ ≤ x <: ⌊ x ⌋ + 1. By using such integer parts, we show that Th(ℝ, 2x) is equivalent to E + Th(ℝ, 21/(H-x3); and we deduce the model completeness of Th(ℝ,2x) from the model completeness ofTh(ℝR,21/(l+x2 )by an argument which differs from Wilkie’s original proof.
Completely random measures
Authors: Kingman J F C
Published: November 2023

It has also the very special property that the random summands on the right of (8.2) are independent random variables. This suggests the concept of a completely random measure on S. This is a random function ct> from the collection of measurable subsets of S into [O, oo] such that, for any collection of disjoint measurable sets A1, A2, A3, ••• , the random variables ct>(Ai) are independent, and The count function of a Poisson process is a completely random measure, but with the additional property that its values are integers. Without this additional condition, a much richer variety of completely random measures can be constructed, but it turns out they all come, in a sense to be made precise, from Poisson processes.
Basins of attraction and the time-dependent approach to obtaining steady-state numerical solutions
Authors: Yee H C
Published: November 2023

The symbiotic relationship between the strong dependence on initial data and the permissibility of spurious stable and unstable asymptotic numerical solutions for commonly used numerical algorithms in computational fluid dynamics (CFD) is illustrated for nonlinear model ordinary differential equations (ODEs) and partial differential equations (PDEs) with known analytic solutions. This is an attempt to understand the global asymptotic behavior of finite difference methods for highly nonlinear and stiff differential equations (DEs). With the aid of a parallel Connection Machine (CM2), the complex behavior and sometimes fractal like structure of the associated numerical basins of attraction of the commonly used time and spatial discretizations are revealed and compared. Here a basin of attraction is a domain of a set of initial conditions whose solution curves (trajectories) all approach the same asymptotic state. Also ¼’e use the term “spurious asymptotic numerical solutions” to mean asymptotic solutions that satisfy the discretized counterparts but do not satisfy the underlying ODEs or PDEs (e.g., nonphysical solutions, nonconvergent solutions or divergent solutions). Asymptotic solutions here include steady-state solutions, periodic solutions, limit cycles, chaos and strange attractors. The results of this investigation can be used as an explanation of the possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.
A general upwind formulation for compressible flows on multi-element meshes
Authors: Dervieux Alain , Fézoui Loula , Leclercq Marie-Pierre , Stoufflet Bruno
Published: November 2023

In the past few years much effort has been devoted to the design of numerical methods applying on unstructured meshes, in order to obtain more complicated methods that can be more easily used, due to mesh generality. This work has been concentrated on mono-element meshes (only one type of element) and particularly on triangles with P1 or Po degrees of freedom. Now, there are strong motivations for considering different elements and multi-element meshes in two possible directions. Firstly, combining triangles and quadrangles can be advantageous in particular for the capture of boundary layers. Secondly, considering higher-order elements is a door to higher-order accuracy in general geometries. For this purpose we start from a Lagrange- Galerkin formulation (Lagrange interpolation, Galerkin variational formulation) and consider upwind extensions. Many upwind methods applying on unstructured meshes are related to Lagrange-Galerkin finite element discretisations. One class of methods, that conserves the finite element philosophy, is the so-called SUPG or Galerkin Least-Square formulation, which has been successfully applied to solve 3-D problems [l]. Recently, distributive schemes ensuring a maximum principle and preserving a linear solution of the advective equation have been studied by several teams (among others [12]). The extension to systems is performed through a multiwave decomposition. Another kind of method, initiated by INRIA and Dassault Aviation, relies on an interpretation of a Galerkin formulation as a Finite Volume centered approximation applied on a dual mesh [2], [3], [4], where the central differences integration is replaced by a Riemann solver [5]. This methodology, which is nowadays widely used, has proved to be reliable as an industrial tool to solve aerodynamic flow problems [6].
A subdomain matching condition for implicit Euler solvers
Authors: Lerat A , Wu Z N
Published: November 2023

In Computational Fluid Dynamics, the multidomain technique provides an efficient way for treating complex geometries with block-structured grids and for using parallel computers. This technique is considered here for solving the steady compressible Euler equations with a time-dependent method.
Surface grid generation with a linkage to geometric generation
Authors: Suzuki Masahiro
Published: November 2023

It has been well recognized that setting grids on the surface is the most time-consuming task in grid generation process and it is a bottle-neck of the Computational Fluid Dynamics (CFD) analysis. This is attributed to a gap between geometric generation and surface grid generation. The purpose of this paper is to present an approach to close this gap. A simple diagnosis of surface grid quality is also proposed.
Grid convergence for adaptive methods
Authors: Warren Gary P , Anderson W Kyle , Thomas James L , Krist Sherrie L
Published: November 2023

Adaptive-grid methods for computational fluid dynamics have gained popularity in recent years due to their potential to provide highly accurate solutions with relatively few grid points. This gain in popularity owes in large part to the difficulty in modeling complex three-dimensional geometries with globally-refined “structured” grids. By using adaptive-grid techniques, it is generally felt that important flow field features, such as shock waves, can be resolved with a high degree of accuracy because of the flexibility to distribute grid points in regions of interest. In addition, since the point distribution enables a large number of points to be concentrated in limited areas, the memory and computational times can be minimized.
Body capturing using adaptive cartesian grids
Authors: Falle S A E G , Giddings J R
Published: November 2023

In recent years there has been considerable interest in using adaptive cartesian grids to compute complex flows (e.g. Berger & Collela 1989; Berger & Leveque 1990; Quirk, J.J. 1990). Such methods are extremely effective for the interior part of the flow, but they suffer from the disadvantage that they do not represent solid surfaces as accurately as body fitted grids. This difficulty can in part be overcome by using a polygonal representation of the body surface (Leveque 1988, Quirk 1992), but it is then impossible to avoid the presence of extremely small cells which lead to very small time steps when using explicit schemes. For inviscid problems this time step restriction can be avoided by allowing waves to pass through more than one cell in a time step (Leveque 1988, Berger & Leveque 1990), merging the small cells with neighbouring large ones (Quirk 1992) or using sub time stepping in the small cells. However, it is not easy to extend these ideas to viscous problems or to more than two dimensions, and it is therefore worth exploring alternative approaches.
Calculation of multiple solutions for the two-dimensional Navier-Stokes equations
Authors: Sobey I J , Mullin T
Published: November 2023

As is the case for many nonlinear sytems of equations, the Navier-Stokes equations may not have a unique solution. There are a great number of papers dealing with bifurcation theory and its application to nonlinear systems yet the study of multiple solutions to the Navier-Stokes equations is only in its infancy [see Benjamin (1978)]. An increasingly important tool in studying such flow fields is computational fluid mechanics. In using CFD to study the innermost structure of fluid flow it has to be recognised that one is attempting to distinguish very fine structural details of flow fields. In addition, many phenomena of interest are inherently unsteady, so it is necessary to use time marching algorithms in order to compute such flows. At present it is difficult to be sure that existing CFD algorithms have the resolution to be used in this way.
Absorbing boundary conditions for acoustic and elastic waves
Authors: Renaut Rosemary A
Published: November 2023

In this paper I review the use of approximate one-way wave equations (OWWES) obtained from rational approximations to as absorbing boundary conditions for acoustic and elastic waves. In particular, I will present a procedure which allows for very easy implementation of high- order boundary conditions for acoustic waves. Similar techniques applied to the elastic wave equation confirm that an Ansatz of Engquist and Majda [2] about the form of a low order boundary condition is correct.
A 2D multiblock method for viscous and inviscid flow computations
Authors: Mensink C , Deconinck H
Published: November 2023

Based on a finite volume discretization of the compressible Euler and Navier-Stokes equations, a multiblock method has been developed in which non overlapping gridblocks are connected in a fully conservative way. The gridblocks may contain different gridline densities with either continuous or discontinuous gridline connections at the block boundaries.
Time-accurate calculations with an implicit Euler /N avier-Stokes solver
Authors: Hoffren Jaakko , Siikonen Timo
Published: November 2023

Euler and Navier-Stokes solvers for steady flows have reached a stage of development where they can be applied to practical problems with a reasonable degree of effort. The required grid resolution and spatial discretization to obtain accurate results in certain types of applications are well known. Among the standard features of the modern codes are the use of a finite-volume formulation and second- or third-order accurate upwind differencing facilitated by the works of Roe and Van Leer. These concepts are also utilized in the thin-layer Navier-Stokes solver developed in the Laboratory of Aerodynamics of the Helsinki University of Technology (Siikonen et al. 1990).
Towards a cell-vertex multi-grid Navier-Stokes algorithm for three-dimensional structured/ unstructured meshes
Authors: Peace A J
Published: November 2023

The problem of predicting the flowfield around complex aircraft configurations continues to receive much interest within the aeronautical research community. To a great extent, the ability of a flow analysis method to examine increasingly complex geometries is paced by the mesh generation capability inherent to the method. In this respect, a number of different strategies have materialised over the past few years for discretising domains around complex configurations. One such strategy, and the one addressed here, is the structured/unstructured approach, whereby the flow-field is composed of regions of structured mesh (hexahedra) and regions of unstructured mesh (tetrahedra). As this approach has its origins in a much used block-structured (multi-block) method, Shaw et al. (1988), the hexahedral cells are contained within structured blocks. The background philosophy behind this approach to mesh generation is given in Shaw et al. (1990).
Handbook of Logic in Computer Science
Authors: Gabbay Dov M
Published: November 2023

The Handbook of Logic in Computer Science is a multi-volume work covering all major areas of application of logic to theoretical computer science. The Handbook comprises six volumes. Each volume contains five or six chapters, giving an in-depth overview of one of the major topics in the field. It is the result of many years of co-operative effort by some of the most eminent frontline researchers in the area. It will no doubt be the standard reference work in logic and theoretical computer science for years to come - essential reading for all those interested in theoretical computer science and logic. volumes cover the background to the subject in terms of mathematical and computational structures. The authors are chosen on an international basis and are leaders in the fields covered. The Handbook is a closely coordinated work which has been under development for the past five years.