bluuebunny/crossref_metadata_2025_split huggingface dataset:
Further ideas
Authors: Huxley M N 
Published: November 2023

Abstract
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.
Some combinatorial aspects of the cover problem for totally categorical theories
Authors: Ivanov A A 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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).
Growth and adaptive models
Authors: Ansell J I , Phillips M J 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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.
A Simple Proof for Non-Monogenesis of the Rings of Integers in some Cyclic Fields
Authors: Nakahara Toru 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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.