bluuebunny/crossref_metadata_2025_split huggingface dataset:
Elliptic Operators and Lie Groups
Authors: Robinson Derek W 
Published: November 2023

Abstract
Elliptic operators arise naturally in several different mathematical settings, notably in the representation theory of Lie groups, the study of evolution equations, and the examination of Riemannian manifolds. This book develops the basic theory of elliptic operators on Lie groups and thereby extends the conventional theory of parabolic evolution equations to a natural non-commutative context. In order to achieve this goal, the author presents a synthesis of ideas from partial differential equations, harmonic analysis, functional analysis, and the theory of Lie groups. He begins by discussing the abstract theory of general operators with complex coefficients before concentrating on the central case of second-order operators with real coefficients. A full discussion of second-order subellilptic operators is also given. Prerequisites are a familiarity with basic semigroup theory, the elementary theory of Lie groups, and a firm grounding in functional analysis as might be gained from the first year of a graduate course.
More geometry for Hering’s 3: SL(2, 13)
Authors: Buekenhout F 
Published: November 2023

Abstract
From there on, we can prove rather smoothly all the other facts mentioned by Hering (1969, 1985) and in particular, his three remarkable spreads. Whilst doing so we get a bit more. For instance, all components of the above spreads are singular subspaces of the symplectic polar space. Next we are adding another contribution to the many ways in which a piece of projective geometry can be used to produce affine geometry. Here, our way to do so, is to use two partial spreads with no common component, such that the unions of the components of each one be equal. We apply this to Hering’s spreads and we find new geometries to add to those he produced already.
Presentations for certain finite quatemionic reflection groups
Authors: Cohen A M 
Published: November 2023

Abstract
Given such a finite linear reflection group, there is a natural and canonical way to isolate a fundamental domain in the reflection space bounded by reflecting hyperplanes. The associated diagram is then obtained by taking for nodes these hyperplanes, joining two nodes if the reflections corresponding to these hyperplanes do not commute and labelling the resulting edge by the order of the product of these two reflections. (This number is of course directly related to the angle between the roots and also the angle between the two hyperplanes.) Conversely, given a diagram, a realization by means of hyperplanes having the right angles leads to a reflection group, namely the group generated by all reflections in these hyperplanes.
Automorphism groups of some generalized quadrangles
Authors: Kantor W M 
Published: November 2023

Abstract
“Most” of the known families of finite generalized quadrangles arise by this procedure (see Kantor (1980); Payne (1990)). It seems likely that Q must be a p-group for some prime p. While this remains open, there has been significant progress (Frohardt (1988); Chen and Frohardt (submitted)).
Internal nuclei of k-sets in finite projective spaces of three dimensions
Authors: Wettl F 
Published: November 2023

Abstract
It is well known that an oval, that is a (q + 1)-arc in an even order plane has a nucleus, that is a point such that every line through this point intersects the oval in exactly one point. Every point of a (q + 2) arc (hyperoval) is a nucleus of the remaining points.
Prime decomposition
Authors: Klingen Norbert 
Published: November 2023

Abstract
In this section we describe the connection between the decomposition of prime ideals in extensions of number fields and group theory. We recall some fundamental facts from algebraic number theory and fix the notation used throughout in this book. We indicate some ideas of the proofs. Complete proofs can be found in any standard textbook on algebraic number theory (e.g. Goldstein [32], Lang [67]). For §§2-4 see also Neukirch [88], ChapterV. a. Prime ideals and residue degrees The arithmetic objects we are dealing with are the prime ideals in the ring of integers in an algebraic number field.
Variations
Authors: Klingen Norbert 
Published: November 2023

Abstract
In this chapter we report on generalizations and refinements of the theory developed in the preceding chapters, as well as on results from related areas which use the same methods or lead to similar group theoretic problems. So this chapter shows the range of questions from number theory, algebra and even from other parts of mathematics which lead to group theoretic questions as discussed in this book. This chapter can be understood as being a guide to the literature and providing some hints for possible further investigations in this area.
The exterior calculus package EXCALC
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D 
Published: November 2023

Abstract
The Cartan calculus of exterior forms is a very powerful and elegant formalism which is increasingly used in classical field theory and especially in general relativity and in gauge theories of gravitation (Thirring (1986), Trautman (1973, 1984)). The REDUCE package EXCALC (Schriifer (1986), Schriifer et al. (1987)) enables one to write programmes that are almost a direct transcription of the exterior form notation.
Analysis of repairable systems
Authors: Ansell J I , Phillips M J 
Published: November 2023

Abstract
This chapter will explore the analysis of data from repairable systems using the models which were discussed in Chapter 5. Knowledge about the process will affect the analysis and this knowledge may have established what is the nature of the underlying process. However, this will not always be the case and sometimes all that will be available will be a list of failure times and possibly also repair times. In some cases this may be aided by knowledge of the failure modes and po.5sibly some knowledge of the system. Obviously the more information available then the better the level of modelling should be. Cox and Lewis (1966) is a good introduction to this area and Ascher and Feingold (1984) is also a useful text on this topic. This chapter concentrates on fitting a variety of models to repairable system data and starts with a few brief comments on the fitting (estimating the parameters) of the models and the assessment of fit. For a more detailed treatment Cox and Snell (1971) is a useful starting point. As with any analysis it is useful to start with simple models and build to more complex models, as was done in Chapter 5. Hence analysts often start by assuming the data may come from a homogeneous Poisson process (HPP) and only reject such a model when it is evident that it is inappropriate. An HPP is a renewal process in which the inter-arrival times have an exponential distribution with the same renewal rate throughout the observed period. This simple model may be inappropriate for many reasons: the underlying distribution may not be exponential, there may be a change in the rate over time, components may be dependent, or the process may not be stationary. Each of these cases will be considered in this chapter as well as other aspects such as the prediction of the future performance of the system. Non-parametric estimators may be a useful guide to decide which model to fit and so estimators of the renewal density function (ROCOF) and the renewal function are presented. It will be assumed in this chapter that the data are the chronologically ordered times taken throughout the observation period at the times the system was repaired.
Case studies in reliability
Authors: Ansell J I , Phillips M J 
Published: November 2023

Abstract
Before embarking on the detail of the three case studies presented in this chapter it is important to recall the role of statistical data analysis within a reliability study. However, to do this it is necessary to recall what the objectives of a reliability study might be, as was discussed in Chapter 1, Section 1.3, and Chapter 9, Section 9.2. The objectives will depend on several factors, such as who has instigated the study and why they have instigated the study. An analysis may be undertaken to establish whether a system has achieved a specific level of performance, or it may be undertaken to assess the likelihood of achieving a successful mission. The identification of where improvements can be made is also a frequent objective. This does not cover all possible aims, but it highlights the differing goals which will often imply different possible analyses. Many of the goals in reliability studies will be achieved without the need for any statistical analysis. Rarely will the statistical analysis play the central role in the study, but where statistical analysis is required then its role may be in assessment, identification, or prediction. The reliability engineer will have an initial objective and this will define the strategy for analysing the data and may also suggest the end point of the analysis. For the case studies considered in this chapter the data consist of life times which have been collected from field studies. There is an important role for laboratory-controlled data, when they are available, but a considerable amount of reliability data is from the field and this chapter concentrates on this situation. The data may be collected on a component or on a system. As well as the lifetimes there may also be associated data on other variables. This accompanying data may qualitatively describe the context or quantitatively describe covariates.
Review of spectral theory
Authors: Eschmeier Jörg 
Published: November 2023

Abstract
This first chapter has an introductory nature; it contains a review of a few classical spectral theories, and then informally explains the main themes of the book. Among these classical concepts are E. Bishop’s ideas on local spectral theory and spectral duality, the theory of decomposable operators of C. Foias, the multidimensional spectral theory of J. L. Taylor, and its homological ramifications. Each of these subjects will be treated in detail in different chapters of the book, and sometimes a topic will appear in various places, treated from different points of view. The bibliographical sources listed in the notes at the end of the present chapter complement our review and, at the same time, they contain some indispensable first references.
Analytic sheaves
Authors: Eschmeier Jörg , Putinar Mihai 
Published: November 2023

Abstract
Algebraic and sheaf-theoretic methods have been used to solve a series of classical problems in the geometry of complex manifolds and in the theory of functions of several complex variables. They have also oriented the research related to several complex variables for a good few decades. Developed in parallel with Grothendieck's theory of schemes, the theory of complex analytic spaces represents nowadays the natural framework and the language in which these algebraic methods are formulated.
Bishop's condition ( 13) and invariant subspaces
Authors: Eschmeier Jörg 
Published: November 2023

Abstract
The main idea of the homological approach to the analytic functional calculus problem is to identify a given commuting system a ε L(X)n of continuous linear Banach space operators with the σ(ℂn)-module structure σ(ℂn) ×X→X→X, (f,x)↦f(a)x determined by a, and then to extend this module structure to the algebra Ma (a)) of all germs of analytic functions defined near the spectrum a (a) of a, applying methods from topological homology theory.
Applications to function theory
Authors: Eschmeier Jörg , Putinar Mihai 
Published: November 2023

Abstract
The abstract notions of quasi-coherence, sub-scalarity, analytic transversality and so on have been related to certain concrete problems which we met in the preceding chapters. These examples offered both a motivation and a validation of the whole theory. The present chapter is intended to bring forward some further examples and applications related mainly to function theory of real or complex variables. This terrain abounds in localization methods and transversality phenomena. Their homological interpretation, based on the preceding chapters, provides a new insight into some old and new results, usually obtained by laborious methods.
Irregularities in the Distribution of Primes
Authors: Friedlander John B 
Published: November 2023

Abstract
In this paper, we survey a body of results, stemming from an idea of Maier [M], which has demonstrated that the distribution of primes in certain basic integer sequences is not as uniform as it was once believed to be. We also make some speculations, both as to further problems that might be amenable to these methods, and as to the nature of some (still apparently) reasonable conjectures to modify those which have been found no longer tenable.
Cubic Inequalities of Additive Type
Authors: Brüdern J , Cook R J 
Published: November 2023

Abstract
The first result on cubic inequalities is due to Davenport and Roth [7]. They proved that if λ1,...,λ8 are real numbers with at least one ratio λi/λj irrational then for any ε &gt; 0 the inequality has infinitely many solutions in integers x1,..., x8. No progress has been made on reducing the number of variables, however the result can be made more precise.
Paulo Ribenboim, at the time of his retirement
Authors: Granville Andrew 
Published: November 2023

Abstract
At this, the third meeting of the Canadian Number Theory Association, we have had the opportunity to enjoy a lot of good mathematics, while celebrating the career of one of Canada’s most distinguished academics. Paulo Ribenboim, author of more than a hundred journal articles and of thirteen books, a Fellow of the Royal Society of Canada, and colleague and advisor to so many mathematicians, is retiring after a career spanning over forty years and three continents. In this article we present a biography of Paulo, together with a sampling of a few of his many interesting results.
The Kummer-Wieferich-Skula Approach to the First Case of Fermat’s Last Theorem
Authors: Granville Andrew 
Published: November 2023

Abstract
In 1909 Wieferich surprisingly deduced, from Kummer’s criteria, that p2 must divide 2P - 2. Soon others deduced that p2 must also divide 3P - 3, 5P -5 and so on; and Frobenius outlined a method to continue proving such criteria.
Images and patterns
Authors: Grenander Ulf 
Published: November 2023

Abstract
In order to analyse and understand a regular phenomenon in nature or in the man-made world we try to describe it in terms of rules, laws, regularities. If we formalize the concept of regularity as was done in the preceding chapter we arrive at the regular structures generally denoted by 𝒞(𝒮), the configuration spaces. A configuration space is a mathematical abstraction that expresses how simple objects, the generators, are combined to form complex systems, the configurations. The combinatory rules leading to 𝒞(𝒮) must now be supplemented by rules of interpretation that express the relation between the configuration, as an abstraction, and what can be observed concretely, seen, heard, measured, and so on.
Abstract biological patterns
Authors: Grenander Ulf 
Published: November 2023

Abstract
Biological growth patterns exhibit so much variability, such dramatic developments, and such a formidable array of possibilities that full understanding of them will probably remain an elusive goal for a long time to come. Attempts to achieve some degree of understanding reach far back; to antiquity as far as descriptive and taxonomic studies are concerned, and to the turn of the century in the case of laboratory work with grafting and surgical experiments (see Note 4.A).
Patterns of collective behaviour
Authors: Grenander Ulf 
Published: November 2023

Abstract
In the dynamics in Chapter 4 the connector graph changed from one subgraph of a lattice to another: the graphs were embedded in a fixed graph, a lattice, which simplifies both the analysis and the algorithms considerably. It is clear, however, that this assumption — all σ’s embedded in a fixed graph — is highly restrictive. We now turn to pattern dynamics where we encounter more essential changes in connector structure, which will lead to some challenging difficulties.
Non-Bayesian pattern inference
Authors: Grenander Ulf 
Published: November 2023

Abstract
In the absence of prior measures on the regular structures we have to use methods of classical statistical inference, such as Neyman-Pearson testing, maximum likelihood and least squares estimation. If the configuration space/image algebra has no detailed structure, say it is just a finite-dimensional Euclidean space with no added relations, then the application of classical statistics need not involve any new ideas. Such cases will not be discussed here where the emphasis is on structured pattern analysis, where the models are intended to incorporate more of the characteristic features of the images under study. We do not get the increased realism of the model free ߝ the price we pay is sometimes considerable in terms of analytical/computational complexity. One such difficulty is caused by the high dimension of the regular structures. This can invalidate standard procedures; maximum likelihood estimation may not even be consistent.
Pattern recognition
Authors: Grenander Ulf 
Published: November 2023

Abstract
The uniform approach to pattern inference developed in Chapters 13-17 enables us to deal with the different tasks - restoration, extrapolation, recognition, etc. - by the same methodology. We have, for example, developed inference algorithms in the same spirit for image restoration, extrapolation, and detection of abnormalities, and shall now show how this can be done for recognition. We have already treated a number of recognition tasks, especially in Chapter 16, but will now study some where the restoration aspect is only secondary.
Creating acceptor functions and connectors
Authors: Grenander Ulf 
Published: November 2023

Abstract
Assume that the connection type Σ contains only graphs without cycles. Then we can sometimes estimate the acceptor functions, or other parameters determining the regularity controlled probability measure, by direct methods; we do not encounter the difficulty just mentioned. To illustrate the technique let us consider a context free language (see Sections 1.3.5 and 8.1.2) assuming for simplicity that it be unambiguous in the sense that each sentence has an essentially unique derivation tree (Note 20.1).
Symmetry
Authors: Woodhouse N M J 
Published: November 2023

Abstract
Symmetry principles play an important part in theoretical physics, both as a means of understanding conservation laws and as computational devices providing quick and elegant solutions to problems that would otherwise be tedious to solve. In quantum theory, their role is especially significant since they not only clarify basic concepts and simplify the formal development of the subject, but they also provide detailed information about physical systems for which no explicit dynamical models exist. However, the methods used to investigate symmetry in quantum systems are often based on abstract group-theoretical techniques, which bypass the canonical formalism. They include no general rules for going over to the classical limit and for directly relating the symmetries of the quantum system to those of a corresponding classical system.
Complex Polarizations
Authors: Woodhouse N M J 
Published: November 2023

Abstract
Real polarizations play an important part in linking classical and quantum mechanics. We shall see that picking a polarization in a classical phase space amounts to choosing a representation in the underlying quantum theory-’representation’ in the sense of a position or momentum space representation. We shall also come across complex representations, which in many ways are easier to handle. An obvious example is the Fock space treatment of the harmonic oscillator, in which the quantum Hilbert space is realized as a space of holomorphic functions Complex representations arise from complex polarizations, which are natural generalizations of real polarizations: one simply relaxes the condition that the Hamiltonian vector fields spanning P should be real.
Quantization
Authors: Woodhouse N M J 
Published: November 2023

Abstract
The problem with prequantization is that the Hilbert space it constructs is too large to represent the phase space of a physically reasonable quantum system. In the first example in §8.4 (a cotangent bundle), the Hilbert space contained wave functions that depended on momentum as well as position; and in the third (a Coadjoint orbit), the group representation was reducible. We have already touched on a possible solution: we should admit only sections of B that are parallel along a polarization. We shall now look in detail at how this works. It should be stressed, however, that the physical justification is not based on general mathematical results (such as the Borel-Weil theorem), but on the way in which the construction works in particular examples. It generalizes and unifies a number of quantization techniques that, in the past, have not appeared to have any obvious connection with each other and that have sometimes seemed overspecialized with applications only to particular physical systems. Also it should be noted that not all syrnplectic manifolds admit polarizations (Gotay 1987).
The Metaplectic Correction
Authors: Woodhouse N M J 
Published: November 2023

Abstract
The half-form bundles are also defined for complex polarizations. By including them in the quantization construction, we arrive at a more useful pairing, which can be applied to a much wider class of nonnegative polarizations. It seems to give the correct physical relationship between the wave functions in the Hilbert spaces of different polarizations. In the case of two Fock spaces, the new pairing is unitary. When P is complex, δp may be nontrivial, and its inclusion in the construction can modify quantization conditions and alter the form of the quantum observables. In the case of the harmonic oscillator, the modification is precisely that needed to get the correct zero-point energy (see §9.2).
Hall bases
Authors: Reutenauer Christophe 
Published: November 2023

Abstract
Hall sets of trees are introduced in Section 4.1. A rewriting system on standard sequences of Hall trees is defined and shown to be confluent, invertible, and always terminating. As a consequence, each word has a unique decreasing factorization into foliages of Hall trees. By interpreting the nodes of a Hall tree as the Lie bracket, one obtains a basis of the free Lie algebra, and the corresponding Poincare-Birkhoff-Witt basis, in Section 4.2. In the last section it is shown that this basis construction gives exactly the same bases as those obtained by elimination in Chapter 0. In the appendix it is shown amongst other things that Hall bases are in some sense optimal.
The action of the symmetric group
Authors: Reutenauer Christophe 
Published: November 2023

Abstract
In Section 8.1 we present the duality between the representations of the symmetric group and the linear group. From this, we quickly deduce the character of the free Lie algebra, already knowing the generating functions of primitive necklaces. This character is induced by a faithful one-dimensional representation of the subgroup generated by a circular permutation (Section 8.2). In Section 8.3 we give the combinatorial interpretation of the multiplicities of irreducible representations in the Lie representation. It is also shown that almost all irreducible representations actually appear. Section 8.4 introduces remarkable Lie polynomials, the Lie idempotents. In the last section, we consider the representations arising from the canonical decomposition of the free associative algebra.