bluuebunny/crossref_metadata_2025 huggingface dataset:
Interpretability
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

We need very little from model theory: just two concepts, and no real results, not even the compactness theorem. The concepts, definability and interpretability, are dealt with in this chapter and the next. The algebraist who has gi en no thought to formal languages will need to look carefully at the material of this section as well as its first few applications subsequently. The motivation comes from algebraic geometry over algebraically closed fields, where definable sets in the technical sense turn out to be the Zariski constructible sets (Zariski open subsets of algebraic varieties) in affine space. The notion of the dimension of an irreducible variety extends to a notion of dimension for constructible sets with little change in its properties, though it becomes somewhat harder to define irreducibility and impossible to define connectivity in this context.
Basic Properties
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

Perhaps the single most important property of groups of finite Morley rank is the descending chain condition on definable subgroups, a rather direct consequence of Axiom A (bearing in mind the discussion of degree in Section 4.2). This leads in the first place to the notion of the connected component of a definable subgroup, and the definable closure of an arbitrary subgroup. The chain condition is discussed in Section 5.1, the connected component is discussed in Section 5.2, and definable closures are discussed in Section 5.5. The main result is Theorem5.12: a connected group has degree 1 (the converse is almost obvious). The other results of Section 5.2 are straightforward but important, and we give 22 exercises (some of them more or less obvious remarks) which the reader should linger over. Exercise 22, page 80, deals with a Schur-Zassenhaus type of theorem and is one of the most interesting exercises, and is not known to hold if Pis assumed to be infinite (or even infinite and central) unless G is solvable (see Theorem 9.32). Section 5.3 prepares for the proof ofZil’ber’s Incomposability Theorem in Section 5.4. This major result proves (simultaneously) the definability and connectedness of certain subgroups of groups of finite Morley rank. In particular if G is connected of finite Morley rank, then its commutator subgroup G’ is definable and connected, and hence by induction the same applies to the higher commutator subgroups aCn) (Corollary 5.30). A similar argument applies to the lower central series on (also in Corollary 5.30). In Section 5.5 we consider the definable closure of an arbitrary subgroup Hof a group G of finite Morley rank, which is by definition the smallest definable subgroup of G containing H. The properties of the definable closure are elementary but useful, particularly Corollary 5.38: the definable closure of a solvable or nilpotent subgroup of given class is again solvable or nilpotent, respectively, of the same class. Again the 13 exercises listed deserve close attention.
Reasoning About Knowledge: A Survey*
Authors: Halpern Joseph Y
Published: November 2023

In this survey, I attempt to identify and describe some of the common threads that tie together work in reasoning about knowledge in such diverse fields as philosophy, economics, linguistics, artificial intelligence, and theoretical computer science, with particular emphasis on work of the past five years, particularly in computer science.
Elementary Geometry
Authors: Roe John
Published: November 2023

The geometry of two- and three-dimensional space has long been studied for its own sake, but its results also underlie modern developments in fields as diverse as linear algebra, quantum physics, and number theory.

This text is a careful introduction to Euclidean geometry that emphasizes its connections with other subjects. Glimpses of more advanced topics in pure mathematics are balanced by a straightforward treatment of the geometry needed for mechanics and classical applied mathematics. The exposition is based on vector methods; an introductory chapter relates these methods to the more classical axiomatic approach.
Axioms for geometry
Authors: Roe John
Published: November 2023

Cut out a large triangle from a big sheet of paper. Not a right-angled triangle or an isosceles triangle or any special kind; just a ‘general’ triangle is what is needed. Label its comers A, B, and C, with AC being the longest side. Arrange the triangle so that AC is horizontal.
Plane geometry
Authors: Roe John
Published: November 2023

In this chapter we will explore in more detail the geometry of a Euclidean space P of dimension 2 or Euclidean plane. We will begin by looking at the concept of orientation, which allows us to distinguish between ‘left-handed’ and ‘right-handed’ coordinate systems.
Conics and other curves
Authors: Roe John
Published: November 2023

In this chapter we will study in more detail some curves in the plane described by ‘locus equations’ of the form f(x, y) = 0, where f is a function of the two variables x and y. Our main attention will be devoted to conics, which are the curves that arise when f(x, y) is a quadratic function of the variables x and y; at the end of the chapter we will also take a look at some examples where f(x, y) is a cubic function of x and y.
Differential geometry of surfaces
Authors: Roe John
Published: November 2023

In this chapter we will study the differential geometry of curved surfaces in three-dimensional space. This subject owes its origin to Euler, but it came to maturity at the hands of Carl Friedrich Gauss (1777-1855), one of the greatest mathematicians of all time. In a memoir entitled General Investigations of Curved Surfaces, published in 1827, Gauss showed that a curved surface could be considered, not just as sitting in some extrinsic three-dimensional space, but as having its own intrinsic geometry; a geometry which, moreover, might be non-Euclidean.
The ruled surface method
Authors: Huxley M N
Published: November 2023

The methods of this chapter are mostly elementary number theory. We want to count the number of integer solutions in bounded regions for certain sets of equations and inequalities. The difficulties are partly algebraic, partly analytic. A system of polynomial equations defines an algebraic variety in affine space. The geometry is dominated by two integers, the dimension and the genus of the variety. The size of the set of integer solutions is usually much smaller than the dimension of the variety would suggest. In the case of projective curves, there are only finitely many integer solutions unless the genus is zero or one (Fallings 1983) (an integer point on a projective curve corresponds to a rational point on an affine curve).
Consecutive minor arcs
Authors: Huxley M N
Published: November 2023

The elementary methods for the First Spacing Problem used the divide-and- conquer method, dividing the solutions of the inequalities into families parametrized by a real number t. The expressions in the different inequalities were related by differentiation with respect to t, and if one inequality held for a long interval of t, then the inequality corresponding to its derivative became sharper.
The Third and Fourth Conditions
Authors: Huxley M N
Published: November 2023

We put together the ideas of the two previous chapters. We want to count coincident pairs among the vectors y constructed from polynomial approximations on the minor arcs. The parameters of the method, M and T, and also H in Chapters 8 and 9 are immutable. The choice of N, the length of the short sums, was made at the outset, and determines R, and so which arcs are major, which are minor. Some fine tuning can be done before applying the large sieve.
Further ideas
Authors: Huxley M N
Published: November 2023

Mathematics delights in general principles and specific examples. The ideas of Bombieri and Iwaniec (1986a) for a specific problem have grown to a general method. A part of mathematics may not die, but it can hibernate, either because all the main results appear to be known, or because the subject has become too complicated. The Bombieri-Iwaniec method is not yet in its final form: we end this book with some suggestions for improving or extending it.
Bases in permutation groups
Authors: Cameron Peter J
Published: November 2023

Let G be a permutation group on a set X. A base for G is a sequence (x1, x2, ... ) of elements of X whose pointwise stabilizer in G is the identity. A base is said to be irredundant if no point in it is fixed by the pointwise stabilizer of its predecessors.
Canonical expansions of countably categorical structures
Authors: Thomas Simon R
Published: November 2023

A permutation group acting on a set Ω is said to be highly homogenous if it acts transitively on the set of n-element subsets of Ω for each n ∈ ℕ. Cameron (1976) has proved the following result.
Some combinatorial aspects of the cover problem for totally categorical theories
Authors: Ivanov A A
Published: November 2023

A finite cover M of a structure N is obtained in the following way. For a finite set F consider a structure M = NU (N ×F) with the natural projection π from N ×F onto N. Some new relations may be added to M but no new structure is induced on N. In this paper we usually assume that N is a strictly minimal countably categorical set. Such covers appear in the investigation of totally categorical theories and the problem of a description of them seems very difficult. We approach this ‘cover problem’ by analysing arities of covers. Mainly, we are interested in what covers can be obtained by adding binary relations to principal finite covers. We show that these covers split and in some natural situations they can be characterized completely.
The small index property and recursively saturated models of Peano arithmetic
Authors: Lascar Daniel
Published: November 2023

The small index property has been proved for several countable structures. See Hodges (1989) and Hodges et al. (1993) for more details. There exist countable ‘.N0-categorical structures which fail to have it (Lascar, 1991), see also p. 52 in this volume. Surprisingly, the case of uncountable saturated structures is easier, and it has been proved by Lascar and Shelah (1993) with the help of GCH. The question of the small index property is closely related to the possibility of recovering a structure from its automorphism group, and in the case of the recursively saturated model of Peano arithmetic, it has been raised by Kaye, Kossak and Kotlarski (Kaye et al., 1991).
Finite-amplitude waves
Authors: Rahman Matiur
Published: November 2023

In the previous chapter the problems of small-amplitude wave theory were discussed. There the free-surface boundary conditions were linearized by assuming that the contribution of second- and higher-order terms was negligible. However, in many engineering applications, the experimental evidence indicates that waves are very nonlinear. For example, in the design of offshore structures, the wave heights of interest are such that the nonlinearities must be considered, requiring that the wave force calculations be based on finite-amplitude wave theory. Therefore, in this chapter we will discuss the wave motion when the wave amplitude is large com pared with the wavelength (see Russell (1844) and Reynolds (1883)).
One-dimensional tidal dynamics
Authors: Rahman Matiur
Published: November 2023

The investigation of long waves is of paramount importance to engineers and scientists in the design of harbours and in obtaining the relevant information about tidal estuaries. Tidal propagation in estuaries is affected greatly by the geometry of the estuary. Long waves propagating into the harbours or the estuary can excite the wave energy into resonance which may cause a great deal of havoc to human beings. It is well known that the tides in the Bay of Fundy are among the highest in the world. Its tidal behaviours is semi-diurnal, with two high tides and two low tides of approximately the same height each day. For the extraction of tidal energy at a suitable dam site of the Bay of Fundy or similar tidal estuaries elsewhere, a knowledge of tidal behaviour is important in terms of con struction operations and subsequent build-up of sediment.
Interpolation and Sampling Theories and Linear Ordinary Boundary Value Problems
Authors: Everitt W N , Nasri-Roudsari G
Published: November 2023

This chapter is devoted to a study of the original Kramer sampling theorem, the link with the interpolation theorem of Shannon-Whittaker, and the connection of these two significant results with boundary value problems associated with linear ordinary differential equations as defined on intervals of the real line. The keystone of this chapter is the connection between these boundary value problems and a generalized form of the original Kramer theorem; this generalization introduces analytic dependence of the kerne1 on the sampling parameter.
Diagram Geometries
Authors: Pasini Antonio
Published: November 2023

Diagram geometry provides a range of techniques that enable an interaction between group theory and geometry. These techniques allow the mathematician to get information on a multi-dimensional geometric object from some knowledge of its bi-dimensional properties. This book introduces these techniques and provides a survey of the development of the subject of diagram geometry. The first three chapters are descriptive; a number of examples are presented, basic concepts are explained, and the reader is introduced to the language of diagram geometries. The theory is developed in the next three chapters and in chapter 7 a number of characterizations are proved. This is continued in later chapters following a survey of more advanced concepts and techniques.
L.Cn-l and more diagrams
Authors: Pasini Antonio
Published: November 2023

Thegeometries belonging to Af.Cn-l will be called Af.Cn-l geometries, for short. Affine polar spaces of rank n, their 2-quotients (in particular, standard quotients, defined in Section 8.4.7) and the Alt(8)-geometry (Ex ercise 7.22) are the only Af.Cn-l geometries we know. Note that an affine polar space can be viewed as an (improper) standard quotient of itself. Thus, when we speak of standard quotients of affine polar spaces, it will be understood that we include affine polar spaces among them.
Mathematical background II
Authors: Mason L J , Woodhouse N M J
Published: November 2023

In this chapter, we shall gather together some basic mathematical results that we shall need to apply the Penrose-Ward transform to integrable systems. The topics covered are: projective spaces and flag manifolds, the geometry of the twistor correspondence, Birkhoff’s factorization theorem, holomorphic bundles, and 2-component spinors and their relation to the twistor correspondence. As in the first part, the intention is not to give a complete or detailed exposition, but rather to define the scope of the tools that we shall need, to introduce notation and terminology, and to record in a convenient form a few basic propositions. Only the first two sections and the statement of Birkhoff factorization theorem are needed as essential background for the construction of the Penros Ward transform in the next chapter.
Reductions of the Penrose-Ward transform
Authors: Mason L J , Woodhouse N M J
Published: November 2023

We showed in the last chapter that the Penrose-Ward transform gives a correspondence between solutions to the ASDYM equation and holomorphic vector bundles over twistor space. We showed how the transform can be used to represent ASD gauge fields by patching matrices: the patching matrix of an ASD connection D generates the connection itself by Birkhoff factorization, and, as we shall see in the next chapter, it embeds D in an infinite family of new solutions, which are related to each other by the commuting flows of the ASDYM hierarchy. Our central interest, however, is not in the ASDYM equations themselves, but in the various integrable systems that can be obtained from them by reduction. We are therefore interested in how the transform can be used to construct solutions that are invariant under subgroups of the conformal group.
The Cohomology of Groups
Authors: Evens Leonard
Published: November 2023

This book presents an account of the theory of the cohomology ring of a finite group. The aim is to present a modern approach from the point of view of homological algebra, and the volume covers themes such as finite generation theorems, the cohomology of wreath products, the norm map, and variety theory. Prerequisites comprise a familiarity with modern algebra and homological algebra as might be gained from introductory graduate courses, though otherwise the book is self-contained. As a result, the book will be useful for those already engaged or commencing in research in this area of mathematics by providing an up-to-date survey of important techniques and their applications to finite group theory.
Additive Number Theory of Polynomials Over a Finite Field
Authors: Effinger Gove W , Hayes David R
Published: November 2023

This volume is a systematic treatment of the additive number theory of polynomials over a finite field, an area possessing fascinating and deep parallels with classical number theory. In providing asymptotic proofs of both the Polynomial Three Primes Problem (an analog of Vinogradov's theorem) and the Polynomial Waring Problem, the book develops the various tools necessary to apply an adelic `circle method' to a wide variety of additive problems in both the polynomial and classical settings. A key to the methods employed here is that the Generalised Riemann Hypothesis is valid in this polynomial setting. The authors presuppose a familiarity with algebra and number theory as might be gained from the first two years of a graduate course, but otherwise the book is self-contained.
On abelian collineation groups of finite projective planes
Authors: Brown J M N
Published: November 2023

Remark. Theorem I is not true for some non-abelian collineation groups. For example, the non-abelian group G = PGL(3, q) acting naturally on PG(2, q3) (or on the Figueroa plane of order q3) has a point orbit P and a line orbit ℒ satisfying: The point orbit (P is the set of all points which are incident with exactly one line of a ^-invariant order q subplane and the line orbit L is the set of lines incident with no point of this sub-plane. See, for example, Brown (1983, §6).
Pointsets in partial geometries
Authors: Clerck F De , Fra A Del , Ghinelli D
Published: November 2023

A (finite) partial geometry of order (s, t) with incidence parameter a (briefly: a pGα(S, t)), is an incidence structure with s + 1 points on a line, t + 1 lines on a point, two points on at most one line, such that for all antiflags (P, I) the number of points on I collinear with P is a constant α ≠ 0. Obviously we must have 1 ≤: α ≤: min (s + 1, t + 1).
Pseudo-symplectic geometries over finite fields of characteristic two
Authors: Liu Y , Wan Z
Published: November 2023

Some known results of symplectic geometries over finite fields will be used below. For the readers’ convenience a brief summary of the symplectic geometries is given at the end of this paper.
Complete arcs in Galois planes of non-square order
Authors: Voloch J F
Published: November 2023

A fundamental quantity in the geometry of projective spaces over finite fields is the cardinality of the second largest complete arc in PG(2, q) denoted by m’(2, q).
Growth and adaptive models
Authors: Ansell J I , Phillips M J
Published: November 2023

The aim of most reliability studies is the improvement of the performance of the system under study. Hence it is reasonable to believe that over a period of time there should or ought to be an improvement in the lifetime of components/systems. Growth models were originally introduced to de scribe such improvement. The early models were constructed on the basis of expected improvement. It should be appreciated that the ‘improvements’ are not always successful and in some cases the system’s performance can be downgraded by the ‘improvements’. Hence the optimism of the growth models should be countered by the pragmatic realization of the possible degradation of a system’s performance over time. The title growth model in reliability has become attached to a set of models which allow for changes in the lifetimes of components over time. These models will include trend models as a special case. In fact trend models are monotonic growth models. Most models assume that growth is monotonic and usually that it is positive with respect to the variable of interest. (The time between failure is increasing, the rate of events decreasing, etc.)
Spectral Decompositions And Analytic Sheaves
Authors: Eschmeier Jörg , Putinar Mihai
Published: November 2023

Rapid developments in multivariable spectral theory have led to important and fascinating results which also have applications in other mathematical disciplines. In this book, classical results from the cohomology theory of Banach algebras, multidimensional spectral theory, and complex analytic geometry have been freshly interpreted using the language of homological algebra. It has also been used to give in sights into new developments in the spectral theory of linear operators. Various concepts from function theory and complex analytic geometry are drawn together and used to give a new approach to concrete spectral computations. The advantages of this approach are illustrated by a variety of examples, unexpected applications, and conceptually new ideas which should stimulate further research.
Square roots of central values of Hecke L-series
Authors: Villegas Fernando Rodriguez , Zagier Don
Published: November 2023

In this paper we will prove this conjecture and analogous results for other grossencharacters. The method will be a modification of the method of [8), where it was shown that the central values of weight one Hecke L-series are essentially the squares of certain sums of values of weight 1/2 theta series at CM points.
Distribution of Small Powers of a Primitive Root
Authors: Montgomery Hugh L
Published: November 2023

From the above we see that the residue classes N are approximately uniformly distributed if N/(p112(logp)2) is large. Presumably more is true. We conjecture that for any f > 0 (1.1)
A Simple Proof for Non-Monogenesis of the Rings of Integers in some Cyclic Fields
Authors: Nakahara Toru
Published: November 2023

Our first method is controlled by the relative trace and the relative norm from Kto the quadratic subfield, where Kis generated by the Gauss sum attached to a biquadratic character. In §5 a second succinct method is shown by using the decomposition law of ideals in the case of cyclic cubic, quartic and sextic fields.
Poincaré Polynomials, Stability Indices and Number of Orderings I
Authors: ƽ Ján Miné
Published: November 2023

In 1985, while working on my thesis under the supervision of Paulo Ribenboim, I observed that one can obtain the sets 0(n) by defining the sets B(n)consisting of Poincare polynomials attached to fields F above with |F/F2| = 2n (see below for details).
General Pattern Theory
Authors: Grenander Ulf
Published: November 2023

The aim of pattern theory is to create mathematical knowledge representations of complex systems, analyse the mathematical properties of the resulting regular structures, and to apply them to practically occuring patterns in nature and the man-made world. Starting from an algebraic formulation of such representations they are studied in terms of their topological, dynamical and probabilistic aspects. Patterns are expressed through their typical behaviour as well as through their variability around their typical form. Employing the representations (regular structures) algorithms are derived for the understanding, recognition, and restoration of observed patterns. The algorithms are investigated through computer experiments.
Some topologies of regular structures
Authors: Grenander Ulf
Published: November 2023

Combinatory regularity is algebraic in character and can be studied from the perspective of partial universal algebra. At the same time it supports other mathematical structures, for example, measures, and, as we shall see now, topologies, see Note 3.A. By this we mean at the moment notions of neighbourhood, convergence, continuity, connectivity, and metrics, but not the topologies that characterize global regularity in terms of the connection type.
Patterns generated from extremum principles
Authors: Grenander Ulf
Published: November 2023

In Chapter 6.4 we encountered pattern dynamics governed by ideas similar to that of mechanics:, a system tends to minimize its energy, or its degree of irregularity. We shall pursue this idea further but now with the emphasis on the final, limiting pattern, see Note 6.A. ‘Energy’ may mean actual energy of a physical system, and a state minimizing it is then called a ground state, or it may represent energy flow, dissipated power or quite different concepts. In a Gibbs measure, where ‘energy’ means the negative of log probability density, a ground state maximizes probability. This will be examined in Chapter 9 and used repeatedly in Part VI as a basis for statistical inference in pattern theory. The study of systems that tend to minimize energy is related to the morphogenetic device in Section 4.3.9 that minimizes irregularity.
Pattern synthesis
Authors: Grenander Ulf
Published: November 2023

Having introduced a regularity controlled probability measure on a configuration space the question naturally arises: can one simulate it? This task, that we shall speak of as pattern synthesis, is important for two reasons. First, during the model-building stage it is imperative to get some intuitive idea of the sort of patterns that are (typically) generated by the measure and to see how their behaviour varies as we change the structure of the model. It is difficult to predict the pattern behaviour directly from the construction of generator space, bond relation, similarity group, etc. To learn about this we shall simulate the measure and display the resulting patterns graphically or otherwise.
Ends and means in pattern analysis
Authors: Grenander Ulf
Published: November 2023

The basic concepts of pattern theory consist of objects and relations. The pattern objects are generators, connectors, connection types, configurations, pure and deformed images, pattern classes. The pattern relations are given in terms of similarity transformations, generator maps, combinatory relations (bond relations, acceptor functions), identification rules, homomorphisms, and deformation mechanisms. The objects are naturally arranged in levels where generators form the lowest level, configurations the next and so on.
Bayesian pattern inference
Authors: Grenander Ulf
Published: November 2023

To realize the tasks in the last chapter we are going to apply Bayesian ideas in Chapters 14–16 and 18, where regularity controlled prior measures have been introduced as in Part IV. If priors are not available we must use non-Bayesian techniques; this will be done in Chapter 17. The philosophy of Bayesian inference, its pros and cons and when it is legitimate to use it, has generated much controversy in the statistical literature. We shall not contribute to this topic, which has already received more attention then it deserves (see Note 14.A). Suffice it to say that when the notion of a prior makes sense and when there is sufficient knowledge about this prior we cannot afford to throw away this subject matter information: a Bayesian treatment is called for. The following chapters are intended for problems when this is the case.
Lagrangian And Hamiltonian Mechanics
Authors: Woodhouse N M J
Published: November 2023

The advantage of the Lagrangian formulation of classical mechanics is that it leads to a simple, direct procedure for writing down the dynamical equations of a mechanical system in arbitrary coordinates. When constraints are involved, this is of considerable practical importance since it allows one to carry out a systematic elimination of the constrained degrees of freedom, and of the constraint forces, in the initial stages of the analysis of a dynamical problem. The procedure is based on the invariance of Lagrange’s equations under general coordinate transformations in configuration space, a fact that is easy to demonstrate by straightforward, but less than transparent calculation.
Elementary Relativistic Systems
Authors: Woodhouse N M J
Published: November 2023

It is often said that spin is a purely quantum concept, with no analogue in classical mechanics; and this is certainly true in the sense that the notion of ‘internal degress of freedom’ is not easy to reconcile with classical particle physics. Nevertheless, it is still possible to construct classical phase spaces for particles with spin from codajoint orbits of the Poincare group. By quantizing these, we can recover the usual wave equations for spinning particles. Geometric methods are essential here because the internal degrees of freedom cannot be separated into configuration and momentum variables. Although this is a formal procedure rather than an attempt to understand internal structure in terms of classical physics, it is useful because it enables us to consider the relationship between the classical and quantum theories of particles with spin within the same general framework as scalar systems. It also leads to a clearer and more direct picture of the connection between the quantum theory of spin and the geometry of space-time than is allowed by the traditional methods of group theory.
Prequantization
Authors: Woodhouse N M J
Published: November 2023

These are Dirac’s quantum conditions (Dirac 1925): they imply that the quantum operators form a Hilbert space representation of the classical observables. In his terms, the Poisson bracket is the classical analogue of the quantum commentator. The quantum conditions do not determine the underlying quantum system uniquely. Nor is every representation of the classical observables a physically reasonable quantization.
Algebraic properties
Authors: Reutenauer Christophe
Published: November 2023

In Section 2.1, we introduce the weak algorithm for noncommutative polynomials, and a similar tool for Lie polynomials. In Section 2.2, we prove the theorem of Shirshov-Witt that each Lie subalgebra of a free Lie algebra is free. In the next section, we show that the automorphism group of ℒ(A) is generated by elementary automorphisms, and give a jacobian-like condition characterizing automorphisms. In the final section, we characterize free sets of Lie polynomials and prove the defect theorem. In the appendix, we discuss uniqueness of rank and restricted Lie algebras in characteristic p.In this chapter, K is a field.
Spherical Splines
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

In this chapter we are concerned with spherical splines. Based on the results of the last chapter we are able to introduce spherical splines in close orientation to the one-dimensional polynomial spline approach. In Section 6.1 we start with a variational characterization of spline interpolation. Section 6.2 shows error estimates, while in Section 6.3 some convergence properties are considered. In Section 6.4 we investigate complete spline bases, while Section 6.5 deals with smoothing by spherical splines.
Gabor and Toeplitz Transforms
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

In many applications such as image and signal analysis, only spectral infor mation about a function (signal) can be observed. Thus we are confronted with the situation that some incoming information, available for a function Fis given as a discrete table of numbers rather than its original form. One can, of course, choose the trial functions in such a way that the countable family is an orthonormal basis of a Hilbert space (for example, the spherical harmonics in £2(0)). The decomposition of Finto these functions is then quite straightforward by taking the Fourier series (that is the orthogonal expansion) of F.However, although Fourier techniques are very powerful, there are some serious deficiencies of the Fourier series approach by means of orthonormal systems: the orthonormalization process and the recovery of the function is in many instances of very undesirable computational complexity, and local changes of the function Fwill affect the whole table of Fourier coefficients, when localization does not occur in space domain (as in the case of spherical harmonics). This shortcoming was already observed by D. GABOR (1946) who proposed to ‘window’ the Fourier integral.
Number of Symbols in Frege Proofs with and without the Deduction Rule
Authors: Bonet* Maria Luisa
Published: November 2023

Abstract

Frege systems with the deduction rule produce at most quadratic speedup over .F’rege systems using as a measure of length the number of symbols in the proof. We study whether that speedup is in reality smaller. We show that the speedup is linear when the Frege proofs are tree-like. Also, two groups of formulas, permutation formulas and transitive closure formulas, that seemed most likely to produce an almost quadratic speedup when using the deduction rule, are shown to produce only log n and log2  n factors respectively.
Poisson Processes
Authors: Kingman J F C
Published: November 2023

In the theory of random processes there are two that are fundamental, and occur over and over again, often in surprising ways. There is a real sense in which the deepest results are concerned with their interplay. One, the Bachelier Wiener model of Brownian motion, has been the subject of many books. The other, the Poisson process, seems at first sight humbler and less worthy of study in its own right. Nearly every book mentions it, but most hurry past to more general point processes or Markov chains. This comparative neglect is ill judged, and stems from a lack of perception of the real importance of the Poisson process. This distortion partly comes about from a restriction to one dimension, while the theory becomes more natural in more general context. This book attempts to redress the balance. It records Kingman's fascination with the beauty and wide applicability of Poisson processes in one or more dimensions. The mathematical theory is powerful, and a few key results often produce surprising consequences.
Recent developments of the Lagrange-Galerkin method
Authors: Priestley Andrew
Published: November 2023

The Lagrange-Galerkin method, which combines the ideas of the method of characteristics with the finite element method, has been popular for solving fluid flow problems for some years now, and has proved very successful in a number of problem areas, such as oil recovery, Navier-Stokes and the shallow water equations, see, for example, Benqué et al. (1982), Bercovier & Pironneau (1982), Douglas & Russell (1982), Lesaint (1977), Morton et al. (1988) & Siili (1988a). The method is very accurate and, because it deals with the advection terms in a Lagrangian fashion, has no CFL limit. This leads, in principle, to unconditionally stability. It can work equally well on unstructured triangular grids or tetrahedral meshes in 3-D.
Iterative solution techniques for the Lagrange-Galerkin method
Authors: Ramage Alison , Wathen Andrew J
Published: November 2023

One area of practical importance in the field of computational fluid dynamics is the solution of convection-dominated diffusion problems for in-compressible viscous flow. The Lagrange-Galerkin method is a numerical technique designed particularly with such problems in mind (see e.g. Morton 1990). The various (weak and direct) forms of the scheme involve using the method of characteristics with a Galerkin finite element approach, combining the power of the latter method with a relatively straightforward finite difference approximation of the Lagrangian material derivative along particle trajectories. If the inner products involved are assumed to be evaluated exactly, the method is unconditionally stable and convergent for a wide range of problems.
Multi-stage Jacobi relaxation as smoother in a multigrid method for steady Euler equations
Authors: Dick Erik , Riemslagh Kris
Published: November 2023

By the use of upwind methods of flux-difference type on steady Euler equations, the first order accurate discretization leads to a set of equations of so-called positive type. This set can be solved by any classical relaxation method in multigrid form. The set of equations generated by the second order accurate discretization does not have this property and as a consequence cannot be solved in the same way. The common approach is then to use defect correction [1, 2, 3]. In this procedure the multigrid method is based on the first order accurate formulation, used as an inner iteration for a second order correction only performed on the finest grid. This procedure proves to work well in many applications. The speed of convergence is however largely determined by the outer iteration and sometimes is found to be rather disappointing, especially when the second order solution is significantly different from the first order solution. It can be expected that if the second order approximation also could be used in the multigrid itself a much better performance could be obtained. A second difficulty is that often convergence cannot be obtained unless a suitable initial flow field is specified, i.e. there is a serious risk of choosing an initial approximation which is out of the attraction region of the iterative method.
Basic Category Theory
Authors: Poigné Axel
Published: November 2023

If asked for a single reason for the attention that category theory, at least as a language, enjoys in some areas of computer science, I wou its attraction stems from being a foundational theory of fun ld guess that ctions which provides a sound basis for (functional) programming and programming logic. If asked for more reasons I would recollect the familiar arguments, namely that category theory
Topology
Authors: Smyth M B
Published: November 2023

Consider a device which outputs a binary sequence. We suppose that the device continues for ever, and also that there is no bound on the time taken to compute each digit. Thus the device may produce an infinite sequence, but may also ‘diverge’, that is, produce some finite sequence and then compute for ever without further output.
Equational Reasoning and Term Rewriting Systems
Authors: Plaisted David A
Published: November 2023

Anequational system is a set of equations. Often we are interested in knowing if an equation follows logically from the given set. For example, given the equationsx + y = y + x, (x + y) + z = x 4-(y 4-z), and — ( —(x +y) + - (x + -y)) = x, we might want to know if the equation — ( —(x +y) + - (x + -y)) = x is a logical consequence. As another example, we might want to know if x *y = y * x in a group in whichx =e for allx. Such systems are of interest for computer scientists as well as mathematicians. Common data structures like lists and stacks can often be described by such sets of equations. Furthermore, systems for mechanizing such proofs on a computer are becoming more and more powerful. In addition, a functional program is essentially a set of equations, typically with higher order functions, and the execution of a program is then a kind of equational reasoning. We will discuss methods of inference that are particularly adapted to equational systems without explicit higher order functions. We will also discuss systems in which the equations may have conditions attached. Next we will consider the use of equational reasoning in general theorem proving programs.
Some Direct Theories of Nonmonotonic Inheritance
Authors: Horty John F
Published: November 2023

Inheritance systems were originally developed within artificial intelligence in response to the practical need for an efficient way of representing and accessing taxonomic information. These systems, along with network representations more generally, were first presented without any semantic analysis at all, or else only with a procedural semantics, according to which the meaning of the representations was supposed to be specified implicitly by the programs operating on them.
Autoepistemic Logic
Authors: Konolige Kurt
Published: November 2023

The subject matter of autoepistemic (AE) logic is the belief set of an agent who has introspective capabilities. It is a modal logic similar in many respects to standard epistemic logics, so that there are modal expressions Lϕ in the language referring to the agent’s beliefs. Where AE logic differs from standard modal systems is in its semantics, that is, the formal account of what an agent’s ‘belief set’ is. To explain the semantics, even informally, requires making the distinction between internal and external logics of belief.
Lambda Calculi with Types
Authors: Barendregt H P
Published: November 2023

The lambda calculus was originally conceived by Church [Church, 193233] as part of a general theory of functions and logic, intended as a foundation for mathematics. Although the full system turned out to be inconsistent, as shown in [Kleene and Rosser, 1935], the subsystem dealing with functions only became a successful model for the computable functions. This system is called now the lambda calculus. Books on this subject e.g. are [Church, 1941], [Curry and Feys, 1958], [Curry et al., 1972], [Barendregt, 1984], [Hindley and Seldin, 1986) and [Krivine, 1990].
Temporal Logic
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

This is the second volume of this respected work in temporal logic. Whereas volume 1 dealt primarily with basic concepts and methods, volume 2 discusses the more applicable aspects of temporal logics. The first four chapters continue the more theoretical presentations in volume 1, covering automata, branching time and labelled deduction. The rest of the book discusses temporal databases, temporal execution and programming, actions and planning.
A General Theory of Structured Consequence Relations
Authors: Gabbay Dov M
Published: November 2023

We want to give an answer to the following question: What is a logical system? Obviously our starting point should be a language which allows us to express declarative statements and we need to give a formal definition of when we accept a reasoning system based on the language as a logical system.
Modal Logic
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators - `like it is necessary' in philosophy, `it is believed' in cognitive science, `it is provable' in mathematics and `it is true after executing a program' in computer science. It is an advanced text which starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantical and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist can use the book as a source of references. For the first time results and methods of many directions in propositional modal logic - from completeness and duality to algorithmic problems - are collected and systematically presented in one volume. Unlike other books, modal logic is treated here as a uniform theory rather than a collection of a few particular systems. It is the only book presenting the theory of superintuitionistic logics.
Bounding pp(μ) when μ < cf( μ) > N0 using ranks normal filters [Sh 386]
Authors: Shelah Saharon
Published: November 2023

Using quite complete (mainly normal) filters to get bounds (even equalities) in Cardinal Arithmetic has a long history. Scott [Sc] proved that for « measurable if GCH holds below k it holds for k by taking ultrapower of the universe, which is still well founded and include P(k). Solovay [Sol] proved that above a compact cardinal.
Cardinal Arithmetic [Sh 400]
Authors: Shelah Saharon
Published: November 2023

We arrived to it for a question for which it gives a satisfying answer. Since the work of Easton we know that if we add (something equivalent to) λcfλ = λ,+ + 2cfλ to the rules of cardinal arithmetic known at this time, then there are essentially no further restrictions. Since the works Devlin and Jensen [DeJ] and Dodd and Jensen [DJ2], we know that this rule holds when there is no 0# and even no inner model with measurables; so if it holds we will say cardinal arithmetic is trivial. It is generally agreed that at least for major problems it is important to find their exact consistency strength. Lately Gitik [Gil] continuing works of Magidor [Mg] and Woodin, show ZFC + 0(k:) = k:++ is an upper bound for consistency strength of “cardinal arithmetic non-trivial”.
The associated Lie ring of a group
Authors: Vaughan-Lee Michael
Published: November 2023

In this chapter we introduce the associated Lie ring of a group. Most of the detailed information we obtain in later chapters about finite groups of prime-power exponent will come from studying their associated Lie rings, and studying the identities which these Lie rings satisfy. First, we establish some of the basic facts we need about the lower central series and the lower exponent-p-central series of a group. We then define the associated Lie ring of a group, describe some of its properties, and go on to determine an explicit basis for the multilinear identities which hold in the associated Lie rings of groups of prime-power exponent.
Kostrikin’s Theorem
Authors: Vaughan-Lee Michael
Published: November 2023

In this chapter we prove:

Kostrikin’s Theorem  Let L be a finitely generated Lie algebra over a field of characteristic p, and suppose that L satisfies the (p−1)-Engel identity xyp−1 = 0. Then L is nilpotent.

This theorem, together with Corollary 2.4.8, shows that the associated Lie rings of groups of exponent pare locally nilpotent, and so solves the restricted Burnside problem for exponent p. We give proofs of Kostrikin’s Theorem for p = 2, 3, 5, and 7 in §3.1, and we give a general proof for p ≥5 in §3.2 and §3.3.
Razmyslov’s Theorem
Authors: Vaughan-Lee Michael
Published: November 2023

In this chapter we prove Razmyslov’s Theorem that if q is a prime power and q≥4 then there exists a non-solvable locally finite group of exponent q. (The theorem for p = 5 was originally proved by Bachmuth et al. [3].) Note that Razmyslov’s Theorem does not hold for prime powers q < 4 since groups of exponent 2 are abelian, and groups of exponent 3 are nilpotent of class at most 3 and hence metabelian.
Groups of exponent four
Authors: Vaughan-Lee Michael
Published: November 2023

In this chapter we prove that groups of exponent 4 are locally finite. This means that the nilpotent quotient algorithm can be used to compute power-commutator presentations for B(r, 4) for small values of r. Such presentations have been obtained for r = 2, 3, 4, 5 and we make extensive use of them to establish commutator identities which hold in groups of exponent 4.
Random Walk In A Random Environment
Authors: Hughes Barry D
Published: November 2023

We now turn from steady-state conduction problems to problems in which a time-evolving random process takes place in a random environment. Our discussion here is restricted to processes on lattices, and in the main to one spatial dimension. Some higher-dimensional problems are addressed briefly in §6.5.3, but the most interesting problems in higher dimensions, those in which percolation phenomena arise, are deferred to Chapter 7. In §6.1-§6.3 we discuss in detail a discrete-time model of random walk in a random environment. A continuous-time model based on a master equation with randomized rate coefficients is discussed briefly in §6.4. Some interesting generalizations of these models are surveyed in §6.5.