bluuebunny/crossref_metadata_2025 huggingface dataset:
Bayesian Estimation of the Grade of Membership Model
Authors: Elena A. Erosheva
Published: November 2023

The Grade of Membership model first appeared in the context of medical diagnosis problem in the 1970s. As a latent structure model for discrete response data, it deals with individual heterogeneity by introducing a set of extreme profiles and individual membership scores for each extreme profile. Existing estimation methods for the GoM model are maximum-likelihood based. Models closely related to the GoM have recently appeared in the genetics and machine learning literatures. We consider a Bayesian estimation approach based on the latent class representation of the GoM model. Augmenting data by the latent class indicators allows us to obtain posterior distribution of the model parameters via a Gibbs sampler, when model hyperparameters are known, or via a Metropolis- Hastings algorithm within the Gibbs, when they are unknown. We illustrate the estimation method on a subset of a disability survey data for the case of two extreme profiles.
Exact solutions of stochastic shear and Bernoulli–Euler beams as benchmark problems
Authors: Elishakoff Isaac , Ren Yongjian
Published: November 2023

A. M. Lyapunov thought that having put the physical initial problem it should be solved mathematically and it would be best if the precise solution was reached—I. V. Andrianov and J. Awrejcewicz (2000). The increasing popularity if the finite element method conceals in it a danger of extreme infatuation with it in damage to analytic methods. Consideration of discrete methods and analytical ones as opposite techniques, that appears in the literature, is not justified. Both methods have characteristic advantages and drawbacks; they mutually complement each other .. .—I. F. Obraztsov (1975).
Variational principles-based FEM for stochastic beams
Authors: Elishakoff Isaac , Ren Yongjian
Published: November 2023

People are divided into two categories: from one side those who see the variational formulation of a physical law the apotheosis of economy of nature; from the other side those who deny them any interest and affirm that they can be formulated only after the physical law of the phenomenon is established—E. Tonti (1969). … It is clear that for any probabilistic modeling of a physical system to be useful, it has to be compatible with the available numerical schemes—R. G. Ghanem and P. D. Spanos (1991c).
This sporting life
Authors: Haigh John
Published: November 2023

Thank goodness the old boxing saw ‘A good big ‘un will always beat a good little ‘un’ is false. Sport would attract far less attention if it were more predictable. Tennis matches turn on the effect of a net cord, or the exact distance a lob from 20 yards will travel. The capricious nature of decisions by officials-whether the ball really crossed the line, why that try was dis allowed-affect outcomes in profound yet random ways. This chapter explores a number of places where an investigation of the effects of chance might lead to a better understanding of the game, or possibly point to better tactics.
Philosophical problems in mental health practice and research
Authors: Fulford K W M , Thornton Tim , Graham George
Published: November 2023

Things have changed a lot since Anthony Quinton, a British philosopher, formerly at Oxford and then a member of the House of Lords, made this observation in a lecture to the Royal Institute of Philosophy (Quinton, 1985). Philosophers are nowadays actively interested in a wide range of topics from mental health practice and research. Conversely, mental health practitioners (including both professionals and users of services) are actively interested in a wide range of philosophical topics. Indeed, the explosive growth in cross-disciplinary contact in the last few years (see Reading Guide) has been as remarkable as the earlier long period of mutual neglect.
Philosophical outputs in mental health practice and research
Authors: Fulford K W M , Thornton Tim , Graham George
Published: November 2023

Thus far in this part we have considered the kinds of problem characteristically tackled by philosophy (Chapters 2 and 3) and philosophical methods (Chapters 4 and 5). In this chapter we will be thinking about the outputs from philosophical work, about what we should expect to get out of doing philosophy. It is important to spend some time on this. Views about the value of philosophy (whether indeed it has any discernible worthwhile outputs at all!) are as diverse as views about philosophical problems and philosophical methods. As we will see, they range from the overoptimistic to the overpes- simistic. So we need to get a balanced perspective appropriate to the particular contingencies of philosophy and mental health.
Phenomenology and psychopathology
Authors: Fulford K W M , Thornton Tim , Graham George
Published: November 2023

This is the first of two chapters in which we will be looking in detail at the philosophical basis of the work of Karl Jaspers, the philosophical father of modern psychopathology. As we saw in Chapter 8, Jaspers envisaged his psychopathology as being founded on phenomenology. In this chapter, therefore, we will look more closely at phenomenology itself, and, in particular, the way it developed in the work of Edmund Husserl. In Chapter 10, we will move on to the second main philosophical influence on Jaspers’ psychopathology, the Methodenstreit, and how Jaspers used this to formulate an approach to psychopathology, which draws on aspects of both the natural and human sciences.
Causes, laws, and reasons in psychiatric aetiology
Authors: Fulford K W M , Thornton Tim , Graham George
Published: November 2023

In each case we have been concerned with issues that, although particularly prominent in psychiatry, are generic to medicine as a whole. In this chapter, by contrast, we will be concerned with a complication to diagnostic assessment that, traditionally at least, has been identified particularly with psychiatry, namely the role that reasons as well as causes may play in how we understand the aetiology of mental disorders.
Knowledge of other minds
Authors: Fulford K W M , Thornton Tim , Graham George
Published: November 2023

These are all questions about the nature of mental phenomena, about their ontology. While they are not straightforwardly empir- ical questions we have looked at the bearing that empirical find- ings, such as the results of brain imaging or theories of mental function from cognitive psychiatry, have on them. However, they have all been questions of ontology, of the nature of what is. By contrast the subject matter of this chapter is the epistemology of mind of how we know about our and others’ mental states.
Histories of the future
Authors: Fulford K W M , Thornton Tim , Graham George
Published: November 2023

There are three kinds of clinical complexity, empirical, valu- ational, and conceptual. Medicine has traditionally been con- cerned mainly with complexity of the first kind, with the empirical challenges of the major pathologies, such as infections, cancer, and heart disease. Partly as a result of its growing success in meeting these challenges, however, late twentieth century medi- cine was marked by the emergence, first of valuational complex- ity, in the form of bioethics, and then, in the closing decade of the century, of conceptual complexity, in the form of a blossoming of new work in the cross-disciplinary field of philosophy of psychiatry.
Understanding the Actions of Others
Authors: Rizzolatti Giacomo
Published: November 2023

There is a set of stimuli that play a fundamental role in our life: the actions of our conspecifics. How do we understand their actions? What are the neurophysiological bases of this ability? It is possible that the capacity to understand the actions of others is based exclusively on the visual analysis of the observed action. However, there is growing evidence that a mechanism underlying this capacity is based on mapping visual information of the observed action on to an internal motor representation of the same action. Here I will discuss evidence for such a mechanism, review its properties, and explore its implication for other cognitive capacities, such as action imitation, understanding intentionality and language.
Core knowledge
Authors: Spelke Elizabeth
Published: November 2023

Does human cognition depend on a set of representational systems that are domain-specific, task-specific, and encapsulated? Studies of human infants and of non-human animals suggest that representations of extended surface layouts, manipulable objects, intentional agents, and approximate numerosities depend on just such modular core knowledge systems. In contrast, studies of human adults, and some studies of trained non-human animals, suggest that representations of space, objects, agents, and numerosity depend on a more flexible cognitive architecture. In this review, I attempt to bridge these findings and suggest that mature human cognitive performance depends in part on the same domain-specific systems found in infants and non-human animals.
Endogenous and Stimulus-driven Mechanisms of Task Control
Authors: Shulman Gordon L , Corbetta Maurizio
Published: November 2023

While cognitive processes can be prepared to carry out pre-specified tasks, they must be sufficiently flexible to respond to new contingencies. We propose that this flexibility is achieved through the interaction of two sets of regions in the human brain. One set is concerned with the endogenous control of tasks and primarily involves dorsolateral prefrontal cortex, intraparietal sulcus (IPs), and frontal eye field (FEF). These regions are active during task preparation, generating signals for selecting task-relevant stimuli and linking them to responses, and maintain these signals during task performance. The other set is recruited by behaviorally important stimuli and involves the temporal–parietal junction (TPJ) and prefrontal cortex.
The imaging of visual attention
Authors: Hillyard Steven A , Russo Francesco Di , Martinez Antigona
Published: November 2023

Imaging studies reported in this book have made substantial advances in delineating the cortical networks responsible for allocating attention to locations and objects in the visual fields. Progress has also been made in specifying the multiple levels of the visual pathways at which incoming sensory information is selected or rejected. To reveal the dynamic mechanisms of attentional control and selection, it is necessary to employ imaging techniques having high temporal resolution (such as event-related potentials and magneto encephalography) as well as techniques that offer anatomical precision (such as functional magnetic resonance imaging).
From viewing of movements to understanding and imitation of other persons’ acts: MEG studies of the human mirror-neuron system
Authors: Hari Riitta , Nishitani Nobuyuki
Published: November 2023

For successful social interactions, humans continuously monitor, and often imitate, motor acts, postures, and gaze of their co-citizens. Recent data suggest that this type of intention reading and action imitation could be supported by ‘mirror neurons’, first identified in the monkey frontal lobe. We have characterized, with whole-scalp magneto encephalographic (MEG) recordings, the human mirror-neuron system (MNS), which seems to consist of several cortical areas activated in a clear temporal sequence. Observation of or facial gestures activated in both hemispheres and within about 250 ms, first the visual cortex, then the superior temporal sulcus (STS) region, the inferior parietal cortex, Broca’s area, and finally the primary motor cortex.
Inference
Authors: Lindsey J K
Published: November 2023

We have now explored some ways of constructing models to help to distinguish between random and systematic variability in an observed sample of data. We have seen how to estimate certain unknown parameters by calculating values from such a sample of data. We also know that such calculated values will generally not be identical to those that we would obtain if we could calculate them for the complete population in which we are interested. It is now time to address several questions concerning how the results that we obtain for our models from a sample can be related to the population as a whole. This is called inference from the observed data to the population.
Normal regression and Anova
Authors: Lindsey J K
Published: November 2023

In Chapter 2, we explored models that allowed the probabilities of the various possible response categories to vary depending on the values of explanatory vari­ables. However, we imposed no specific constraints as to the shape of the prob­ability distributions under this different conditions ( except for order in Section 2.3.4) so that a large number of parameters was often involved. In Chapter 4, we looked at how the forms of probability distributions could be simplified by adopting some appropriate density function. Two major advantages of this pro­cedure were that an appropriate density function should be informative about the data generating mechanism and that the number of parameters to estimate was reduced.
Ecological morphology and anatomy
Authors: Gibson David J
Published: November 2023

The common possession of a number of characteristic morphological features makes grasses readily recognizable to even the lay person— although, as noted in Chapter 2, there is often confusion with other similar families such as the rushes (Juncaceae) and sedges (Cyperaceae). In this chapter the morphology and anatomy of a grass plant is described, with mention where appropriate of the ecological relevance and importance of the different features. A number of other sources can be consulted for additional details such as Metcalf (1960), Langer (1972), Chapman (1996), Gould and Shaw (1983), or Renvoize (2002) as well as electronic databases (see §2.2) including Watson and Dallwitz (1992 onwards). There is considerable diversity in form and function that relates to the evolutionary adaptation of grasses to their environment. In general, the trends in grass evolution reflect a series of adaptive reductions in size, complexity, and number—albeit with notable exceptions and a number of reversals (Stebbins 1982).
Physiology
Authors: Gibson David J
Published: November 2023

Grasses are remarkable in their combination of several unique physiological features. These include the possession in more taxa than any other group of the advanced C4 photosynthetic pathway (§ 4.1), highly nutritious forage (§ 4.2), the production of economically important chemicals including starches, sugars, and other carbohydrates (§ 4.3), and the possession of large amounts of silica in their tissue (§ 4.4). All of these physiological features, and others, fundamentally affect the ecology of grasses (§ 4.5), and are described in this chapter.
An act of desperation
Authors: Baggott Jim
Published: November 2023

A scientist in the late nineteenth century could be forgiven for believing that the major elements of physics were built on unshakeable foundations and effectively established for all time. The efforts of generations of scientists, philosophers, and mathematicians had culminated in Isaac Newton’s grand synthesis in the late seventeenth century. Newton’s work had been shaped by a further 200 years of theoretical and experimental science into a marvellous structure that we now call classical physics. This physics appeared to explain almost every aspect of the physical world: the interplay of force and motion in the dynamics of moving objects, thermodynamics, optics, electricity, magnetism, and gravitation. Its scope was vast: from the objects of everyday experience on earth to the furthest reaches of the visible universe. So closely did theory agree with and explain experimental observations that there could be no doubt about its basic correctness —its essential ‘truth’. Admittedly, there were a few remaining problems, but these seemed to be trivial compared with the fundamentals — a matter of dotting a few is and crossing some ts.
Quantum measurement
Authors: Baggott Jim
Published: November 2023

Measurement lies at the core of the quantum formalism. This may seem a statement of the intuitively obvious for any theoretical structure with a pretension to predict the outcomes of observations made or experiments performed on objects in the real world. But classical mechanics is a theorys of the physical properties of objects and their behaviour when exposed to different kinds of forces, and there is no particularly significant role for measurement other than a passive role involving the recording of those properties or behaviour. In contrast, measurement in quantum theory takes on a far deeper, almost ominous, significance
Plantation organization and structure
Authors: Evans Julian , Turnbull John W
Published: November 2023

In Chapter 1 the words ‘plantation’ or ‘planted forest’ were defined and described in outline. But it is nec-essary to examine more closely the organization and structure of plantations and planted forest as background to their development and management. Although most of this chapter is concerned with industrial forest plantations, much of it also applies to small-scale woodlots, though the relative importance of some matters may differ, for example, species composition, age-class structure, stocking, design, and shape of the plantation.
Seed collection, supply, and storage
Authors: Evans Julian , Turnbull John W
Published: November 2023

Growing seedlings in a forest nursery is the main way of raising planting stock in the tropics, though increasingly some plantation species are propagated vegetatively. Large quantities of seeds are needed each year for industrial plantations and smaller-scale planting by farmers, villagers, smallholders, and other land managers. There is abundant evidence from around the world (e.g. Jones 1994) to show that the availability of good quality seed of the best adapted species is one of the most critical factors in ensuring healthy, productive, and profitable plantations.
Rotations, regeneration, and coppice
Authors: Evans Julian , Turnbull John W
Published: November 2023

These questions about regeneration cannot be decided in isolation. For example, if natural regeneration is planned then the rotation must be long enough for trees to produce seed abundantly. The main methods of regeneration used in tropical plantations, that is, clear-felling and replanting, and coppice systems, were briefly discussed in Chapter 4. This chapter considers them more fully, along with brief mention of natural regeneration, but, first, two important points are restated.
Ecological factors and sustainability of plantations
Authors: Evans Julian , Turnbull John W
Published: November 2023

In this final chapter three issues are addressed that affect planted forests to a greater or lesser extent. They are specifically included because debate and sometimes controversy surround them, namely, biodiversity in plantations compared with natural forest formations, the eucalypt controversy, and plantation sustainability. Any casual examination of blocks of planted trees consisting of one species, and often of one age, compared with natural forests of many species and ages shows they are less rich in species. Obviously, and virtually by definition, this is true of tree species, but also applies to associated flora and fauna. This has led to claims that plantations are biological deserts.
From genealogies to sequences
Authors: Hein Jotun , Schierup Mikkel H , Wiuf Carsten
Published: November 2023

In the previous chapter we discussed how genes or sequences are related in a population through their common ancestry. However, to model real data a model of how mutations cause changes in the DNA is necessary. To meet this several mutation models have been developed. Historically, the infinite alleles model appeared first (Kimura and Crow 1964), followed by the infinite sites model (Kimura 1969), and the finite sites model (Jukes and Cantor 1969). This development illustrates an increased focus towards analysis of nucleotide sequence data sets, but also a development towards increasing complexity of analysis.

In this chapter we first introduce the three different types of mutation models, then a mathematical framework for working with probabilities of a sample configuration is discussed. We will mainly focus on the infinite alleles and infinite sites models and relate these models to the underlying genealogy of a sample. Mutations are assumed to be selectively neutral. This has the desirable effect that the mutation process can be separated from the genealogical process, because, in the absence of selection, the mutational process and the transmission of genes from one generation to the next are independent processes (genes have the same probability of transmission whatever their type). Thus, a sample configuration for n genes can be simulated using a two step procedure: (1) simulate the genealogy of n genes; (2) add mutations to the genealogy according to the chosen mutation model.
Human evolution
Authors: Hein Jotun , Schierup Mikkel H , Wiuf Carsten
Published: November 2023

Humans are tremendously interested in human evolution and this topic gets more attention than the evolution of any other species. This is so for several reasons. Beyond the obvious, that we are interested in our own evolution, human evolution is tied up with the interpretation and understanding of diseases with a genetic component and is influenced by human demographic and cultural history of which much is known. For other species than our own less is known about demography and culture.

Until recently, longer term human evolution (thousands to millions of years) has mainly been investigated by comparisons of dated fossils and archaeology. Such analysis has brought much understanding of the time of origin of anatomically modern humans, the major migrations and colonisa- tion of the earth, and knowledge about demography. These studies have the advantage that they often use traits of direct interest such as brain volume, upright posture, and pathologies.
Convex polytopes
Authors: Webster Roger
Published: November 2023

The concept of a convex set is a simple one. A set in space is convex if whenever it contains two points, it also contains the line segment joining them. Elementary geometry abounds in convex sets: ellipses, triangles, parallelograms, balls, halfspaces, and cubes are convex. An annulus, a crescent, and the vertex set of a cube are examples of nonconvex sets. Convex and non-convex sets are illustrated in Fig. 2.1.
Classification and Construction of Buildings
Authors: Ronan Mark
Published: November 2023

The purpose of this article is to give some feeling for what a building is, to discuss their classification, their construction, and something of their local structure. For more details two recent books, [Br] and [R2], are available, both published in 1989, plus a survey article [R3] which contains an introductory treatment of the subject.As is well known, the theory of buildings was introduced, and has been largely developed, by Jacques Tits. Its original purpose was to provide an understanding of the analogues of the simple Lie groups, and particularly the exceptional groups, over an arbitrary field, and in this respect it has been brilliantly successful. The theory of spherical buildings and their classification [T2] was the result, but things by no means stop there. Work of lwahori and Matsumoto [IM] gave us affine buildings; the general theory of affine buildings was then developed by Bruhat and Tits [BT], and a classification was accomplished by Tits [T7].
Groups Acting Transitively on Locally Finite Classical Tits Chamber Systems
Authors: Meixner Thomas
Published: November 2023

Tits chamber systems admitting a transitive group of automorphisms have seen considerable interest in the last six or seven years. As a general reference I want to mention the surveys (lectures held at the Como conference in 1984) by Kantor ([61) and Timmesfeld ([28]). Some notions have been developed, some methods such as the “amalgam method” have proved to be very powerful instruments since then; different notions used, however, cause people to use different sets of hypotheses, and sometimes the interested reader might have trouble to see how the numerous papers written fit together and whether the results achieved give a completeclassification of the objects one is interested in.The purpose of this survey is to state the classification theorem that has been achieved since Como, to give the list of “types” allowed by the theorem together with explicit examples found by many authors, finally describe the systems of subgroups used in the literature with their various relationships. Thereby, I will have to repeat parts of Timmesfeld’s survey, but will do so in order to be as clear as possible.
Generalized Polygons and s-Transitive Graphs
Authors: Weiss Richard
Published: November 2023

Stabilizer Gacts transitively on the set of s-paths beginning at x.If, in addition, Gacts transitively on the vertex set off, and hence on the set of alls-paths in r, then f is called s-transitivewith respect to G.We will give here a brief survey of various efforts to extend and apply this and some related (but historically independent) results of Tits. As these developments include the entire theory of amalgams opened up byD. Goldschmidt’s seminal paper [8] on locally s-transitive trivalent graphs, our choice of results will, of necessity, be very selective. In general, we will describe results which are of a more graph theoretical nature. The reader is referred to (22] for an earlier report on s-transitive graphs.
Synthetic differential geometry
Authors: McLarty Colin
Published: November 2023

The synthetic differential geometry axioms (SDG) describe a topos in which each object has a differentiable structure, each arrow has a derivative, and the basic rules of calculus are simple calculations with infinitesimals. For this chapter we assume that we have a particular topos Spaces that satisfies the axioms. We refer to objects of Spaces as spaces, to its arrows as maps, and to global elements p: 1→.M aspoints of the space M.
Geometric Stability Theory
Authors: Pillay Anand
Published: November 2023

This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.
Stability theory
Authors: Pillay Anand
Published: November 2023

In this chapter, I survey the stability-theoretic results and notions required for the rest of the book. I include forking, canonical bases, ranks, orthogonality, and stable group theory. I give selective proofs. Hopefully this chapter will also provide a point of entry into general stability theory for the beginner, and thus I include a fairly thorough treatment of forking. I will be assuming knowledge of the basic ingredients of first-order model theory (languages, structures, elementary substructures, compactness, types, saturation, etc.). Such background can be found in Chang and Keisler’s book [CK] for example, and also in Hodges recent book [Ho], but notation will be consistent with my earlier book [Pl].Strictly speaking, a full grasp of stability theory is not required for understanding the ‘finite rank’ part of the theory, covered in Chapters 2, 3, 5, 6, and part of Chapter 4. For much of this material, simply dimension and canonical bases in strongly minimal sets or minimal sets are all that the reader has to understand. So the reader who so wishes can begin at Chapter 2 and refer back to various sections of Chapter 1 (on minimal sets, ranks, etc.) when necessary. On the other hand, general stability provides a unifying conceptual framework and language for the various topics covered in the book, as well as being an essential feature in Chapter 4, 7, and 8.
Quasifinite axiomatizability
Authors: Pillay Anand
Published: November 2023

We saw in Chapter 2 that totally categorical theories are not finitely axiomatizable. A slight extension of the proof yields non-finite axiomatizability of ω-categorical, ω-stable theories. At some point it was asked whether the only obstruction to the finite axiomatizability of, say, totally categorical theories was the non-finite expressibility of ‘there are infinitely many elements’, that is whether totally categorical theories could be finitely axiomatized modulo the ‘axiom of infinity’. In this chapter we see that this is true.
Solvable Groups
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

The structure theory of solvable groups of finite Morley rank is in a quite satisfactory state, though by no means as detailed as the theory available in the algebraic case [189). We obtain the conjugacy of Sylow subgroups for arbitrary primes (whereas the general theory of the next chapter works only for the prime 2) as well as the extensions of the Sylow theory due to Philip Hall (Section 9.9). Section 9.1 contains Zil’ber’s very useful technical result on the structure of certain actions of one abelian group on another, which leads to the theorem in Section 9.2 that the derived subgroup of a connected solvable group G of finite Morley rank is nilpotent. In the algebraic case there is also a representation of G as a semidirect product of an abelian group T acting on a nilpotent subgroup U. We would like such a result in our case, with at least U definable, but this problem is open. We discuss it in general in Section 9.3 and in the case of connected solvable groups of class 2 in Section 9.4. Specializing further in Section 9.5 to the case of rank 2, we obtain a detailed structure theory in the nonnilpotent case, and we make some further comments on the nilpotent case, motivated by the observation that in the study of solvable connected groups of finite Morley rank, nilpotent groups often arise in practice as the commutator subgroup of a given group. One candidate for the subgroup U above is the connected component of the Fitting subgroup, F (G)0 In Section 9.6 we show that the quotient G/ F(G)0 is divisible abelian (that it is abelian is already contained in the result of Section 9.2).
2-Sylow Theory
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

In this chapter we prove the conjugacy of Sylow 2-subgroups of groups of finite Morley rank and give some useful applications of this which are analogs of results on finite groups that play a major role in the classification of the finite simple groups (and which we apply subsequently in the same fashion). Since we do not have the ability to carry out counting arguments, the proof proceeds mainly by induction on rank. Sections 10.1 and 10.2 contain some general properties of involutions and their actions on invariant abelian subgroups, as well as an analog of the Brauer-Fowler theorem (Theorem 10.3). Section 10.3 contains the major result: Sylow 2-subgroups are conjugate. They are usually not definable, hence the notion of rank does not apply directly to them.
Geometries
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

This chapter contains results on various geometric structures related to groups of finite Morley rank. We will first look into some incidence geometries: the generalized n-gons and their flag-transitive automorphism groups. The generalized n-gons have been introduced by Tits [343] in 1959 in connection with certain group theoretic problems, and they generalize the projective planes. In Subsection 13.2, we will see that a bad group of Morley rank 3-which is not known to exist-acts sharply flag-transitively on a pseudoplane that looks very much like a projective plane (Theorem 13.16).
CN and CIT-Groups
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

In this chapter we will study two classes of groups: the CN and CIT-groups of finite Morley rank. CIT-groups are groups with involution where the centralizer of every involution is a 2-subgroup. CN-groups are groups where every centralizer is nilpotent. Note that simple bad groups are CN-groups. For this reason we have to assume that a CN-group has involutions. The classification of CN and CIT-groups in the context of finite groups uses character theory. Character theory is not available to us. We compensate this lack by computing in the group itself. Some of the computations that we need have been done in Subsection 11.5.5. In this chapter we will show that most of the CIT and CN-groups are split Zassenhaus groups and we will then apply the results of Subsection 11.5.5.
Effective representation systems
Authors: Smullyan Raymond M
Published: November 2023

In this chapter we apply the results of the last chapter to the representation systems of Chapter 4. For any positive n we define r  n+1 (x,y  1, ... ,y  n) to be the Gödel number of Ex(y  1,... ,y  n). We call Z an effective representation system if: (1) for each positive n, the function r  n+1(x,y  1, ... ,y  n) is recursive; (2) the set of Gödel numbers of the sentences is recursive, and for each n, the set of Gödel numbers of predicates of degree n is recursive. Throughout this chapter Z will be assumed to be an effective representation system. We let r(x,y) be r  2(x,y) — thus r(x,y) is the Gödel number of Ex(y) and the diagonal function d(x) is r(x,x).
Synchronization and pairing functions
Authors: Smullyan Raymond M
Published: November 2023

In this chapter we consider a totally different approach to double and multiple fixed point properties. This approach is an outgrowth of the original form of the double recursion theorem[28] which involves a pairing function J(x, y) and its inverse functions K(x), L(x) for its very statement. We will deal with this in Part III of this chapter.
Formal combinatory logic
Authors: Smullyan Raymond M
Published: November 2023

In preparation for the fixed point theorems of the next chapter, we must turn to some basics of axiomatic combinatory logic.
Transitivity and orbits
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

For any field F the linear group GL(n, F) is transitive on F  n\{0}. For, given a non-zero vector u in F there certainly exists an invertible n x n matrix A over F, whose first row is u and then.
Applications of Sampling Theory to Combinatorial Analysis, Stirling Numbers, Special Functions and the Riemann Zeta Function
Authors: Butzer P L , Hauss M
Published: November 2023

The Shannon sampling theorem together with its version for not necessarily band-limited functions is essentially equivalent (in the sense that each can be deduced from any of the others by elementary means) to at least five fundamental theorems in four different fields of mathematics. namely to the Poisson summation formula of Fourier analysis, to a particular form of Cauchy’s integral formula of complex function theory, to the Euler-Maclaurin and Abel-Plana summation formulae of numerical analysis as well as to the celebrated functional equation of the Riemann zeta function of analytical number theory. Recall (Higgins 1996, pp. 90-96) and check Section 1.6 below.
Mathematical background I
Authors: Mason L J , Woodhouse N M J
Published: November 2023

In this chapter, we summarize the mathematical background to the Yang-Mills equations. In §2.l we give an informal overview of gauge theories and their relation to the anti-self-duality condition. In §2.2-§2.4 we give brief notes on the geometry of real and complex space-time, and on conformal transformations, followed in §2.5-§2.6 by a more formal discussion of bundles, connections, curvature and the Yang-Mills equations. These topics are covered more fully by Ward and Wells (1990).
Reduction to three dimensions
Authors: Mason L J , Woodhouse N M J
Published: November 2023

The ASD condition reduces to a system of equations in three dimensions when we take the quotient of a neighbourhood in space-time by the action of a one-dimensional group generated by a single conformal Killing vector. The general features of the various cases are determined by the properties of the generator: whether or not it is null, whether or not it is hypersurface orthogonal, and how it behaves near its zeros (the fixed points of the action). We shall see two significant phenomena in the case of a null translation. First, the invariants of the Higgs field are constant on the a-planes orthogonal to the generator: these ‘first integrals’ classify further reductions. Second, we find that the reduced system acquires an unexpected infinite-dimensional point symmetry group.
Twistor construction of hierarchies
Authors: Mason L J , Woodhouse N M J
Published: November 2023

In Chapter 8, we saw that a given solution to the ASDYM equation could be embedded in an infinite family of new solutions by moving it along the commuting flows of the ASDYM hierarchy. The lowest level flows in the hierarchy are the translations in space-time, from which the higher flows are generated by the recursion operator.
Finite p-groups acting uniserially
Authors: Leedharn-Green C R , McKay S
Published: November 2023

This chapter concentrates on the idea of a finite p-group G acting uniserially on a finite p-group N; that is to say, G acts on N, and the G-invariant subgroups of N are linearly ordered, there being precisely one of index pi in N for each i ≥ 1 with pi ≥ INI. This situation arises naturally in a finite p-group G of coclass r as the action of G on all but r of the sections of any normal series of G must be uniserial: see Lemma 4.1.15.
The proof of Conjecture A using powerful p-groups
Authors: Leedharn-Green C R , McKay S
Published: November 2023

At the heart of the basic techniques in p-group theory is the interplay between the commutators and the power structure. In particular, various classes of groups have been introduced in which some control is exerted over this interplay. The classic case is the class of regular p-groups introduced by P. Hall. These are described briefly in Section 1.2. More recently the class of powerful p-groups was introduced by A. Mann. Both classes contain the class of abelian p-groups, so in each case the properties obtained are to some extent generalizations of results for abelian groups.
Invariants Of Smooth Four-Manifolds
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

We turn now to the question which formed the second main thread of the topological discussion in Chapter 1, namely the question of distinguishing smooth four-manifolds having the same classical invariants. Our strategy is to define new invariants using the ASD moduli spaces. The ASD equations are not defined until a Riemannian metric g (or rather, a conformal class) is chosen; the space of solutions-the moduli spaces we have been studying-reflect accordingly many properties of the metric. In order to define differential-topological invariants, we must extract some piece of information from the moduli space which is insensitive to a change in the metric and therefore depends only on the underlying manifold.
Structure of H
g
Authors: Dwork Bernard
Published: November 2023

We consider the situation of Chapters 2 and 3, in particular hypothesis (2.4) but we shall not use hypotheses (2.5) and (2.6) until the second part of this chapter (§ 6.5 onwards). It will be shown later Lemma 9.1.1 that W^a is finite dimensional (and hence its adjoint space ℋ^a is dual) but this will not be used immediately. Our object is to find necessary and sufficient conditions under which for given u∈H^0 the map is an injective map of K^a into γ^−∘Xu The necessity of our condition is not established.
Generic Case
Authors: Dwork Bernard
Published: November 2023

In this section we return to the situations of Chapter 1 and of § 3.1.A. We ignore Jr in the treatment of this section.
L-Functions and Representation Theory of p-Adic Groups
Authors: Shahidi Freydoon
Published: November 2023

The modern theory of automorphic forms as envisioned by Langlands (28, 30] gives an equal weight to representation theory of p-adic reductive groups as that of real groups. While representation theory of real groups is near perfection, a similar statement cannot be made about p-adic groups and in fact there is an abundance of unsolved problems of extreme depth, beauty, and importance that must be answered. The work of Harish-Chandra [11, 45] has already proved that a large part of the theory from real groups carries over, fairly precisely, to the p-adic case. Nothing like this can be said of their discrete series representations and in fact to this date our understanding of them remains poor, except for GLn and certain other classical groups (see the next paragraph).
Weak Forms of Amenability for Split Rank 1 p-Adic Groups
Authors: Valette Alain
Published: November 2023

Let F be a non-discrete locally compact field and let G be the group of F-rational points of a simple algebraic group defined over F. Let G have positive split rank, so that G is not compact in the locally compact topology coming from F. For example, G = SLn(F) when n ⩾ 2. It is then a classical fact that G is not amenable. We recall the proof below. However, if F is a local field and G has split rank 1 (for example, G = SL2(Q p ), or SL2(D), where D is a central division algebra over Q p ), then weak forms of amenability do hold. The aim of this paper is to survey weak forms of amenability.
So(3)-Connections
Authors: Petrie Ted , Randall John
Published: November 2023

We begin by recalling some notation from Chapter 2. Let E be a G-vector bundle, where G = U, O, or SO, and let dE be a G-connection on E, that is, a Hermitian (respectively orthogonal) connection on E when G = U (respectively G = O or SO).
The Index Of The Fundamental Complex
Authors: Petrie Ted , Randall John
Published: November 2023

In this chapter, we develop the background needed to define elliptic complexes and state the Atiyah-Singer index theorem. We introduce the fundamental complex associated with a self-dual connection on an SO-vector bundle over a compact oriented 4-manifold M.
A Motivated Introduction to the Langlands Program
Authors: Murty M Ram
Published: November 2023

This paper is partially expository and is intended as an introduction to the Langlands program for number theorists. The new theorem in the paper is that automorphic induction map for Hecke characters implies both the Artin conjecture and the Langlands reciprocity law. (See below for definitions.) In the last section, we describe some recent work with K. Murty [18] that applies the theory of base change to elliptic curves. The paper is not an exhaustive survey. We have tried to use some classical problems of number theory as motivation for discussion. For instance, we concentrate on GLnthough the functoriality conjecture predicts that this is not a limitation. Nevertheless, our discussion is sufficiently motivated from the number theoretic point of view that a non-specialist in the field can appreciate the depth and profundity of these ideas.
Ordered Factorizations for Integers and Arithmetical Semigroups
Authors: Knopfmacher A , Knopfmacher J , Warlimont R
Published: November 2023

The classical problems of “factorisatio numerorum” concern (i) the total number of ordered factorizations of a natural number n > 1 into products of natural numbers larger than 1, and (ii) the corresponding total number of unordered factorizations of the same type. Results of this nature can be found, for example, in [l], [7], [12] and the references therein. Recently, in [8], the present authors considered the analogous problems within the context of additive arithmetical semigroups as treated in the monographs [10], [11].
Projection fro1n an Algebraic Quadratic Form to Rational Quadratic Forms
Authors: Cohn Harvey
Published: November 2023

0. Introduction. Class field theory, in its historical ongms, first centered on the study of conditions on a rational prime p which lead to its representation by a positive definite quadratic form in rational integers (0.1) p = ax2+bxy+cy2, (d=b2-4ac<0), where (a, b,c) = 1, but the discriminant d need not be fundamental.
Problèmes de Favard et de Lehmer Generalisés
Authors: Langevin Michel
Published: November 2023

In this work, we describe how to extend these results in the case of any number field and how to introduce the problem in the context of the generalized Lehmer’s problem ; moreover, we give some improvements in the case of self-inversive polynomials.
Polynomials
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

In this chapter we are concerned with the theory of homogeneous harmonic polynomials in Euclidean space R3. The main result is the addition theorem. When restricted to the unit sphere this theorem will play later on a central role in the theory of spherical radial basis functions.
Tensorial Approximation Methods
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

Following the formalism established for tensor spherical harmonics we are now able to generalize our approximation methods directly to the tensorial framework. This can be done quite similarly to the method indicated in Chapter 13 for vector fields, which is why we only state the keypoints here.
Model Theory and Computer Science: An Appetizer
Authors: Makowsky J A
Published: November 2023

The purpose of this chapter is to give an account of those aspects of model theory which we think are relevant to theoretical computer science. We start with a general outline of the evolution of model theory, which will serve as an exposition of the major themes. In the subsequent section we shall elaborate some of these themes and put them into the context of theoretical computer science.
Basic Modal Logic
Authors: Fitting Melvin
Published: November 2023

Modal logics are logics of qualified truth.Necessary, obligatory, true after an action (such as running a computer program),known, knowable, believed, provable, from now on, so far, since anduntil are qualifiers with similar formal characteristics. But while they are related, these qualifiers differ markedly from each other.Necessity can have different properties depending on whether physical necessity, logical necessity, or something else is being considered. Even if this has been decided, can we have a sentence whose truth is necessary, without that fact in turn being necessary? Briefly, if some truth is necessary, is it necessarily necessary? What if a sentence is not necessarily true; must that be a necessary truth? In every-day life we are not in the habit of asking such questions, and so may have little feeling for appropriate answers.Known andbelieved differ in a basic principle: what is known must be true, but what is believed need not be. Time can be thought of as discrete or continuous, the future as fixed or not, the past as determined or not, so qualifiers likeuntil can be given varied interpretations. In addition to variations motivated by considerations like these, large numbers of modal logics have been created for purely formal purposes: their mathematical properties are of interest for some reason or other. Clearly there are a great many modal logics and it is impossible to cover them all in even passing detail.
Unification Theory
Authors: Baader Franz , Siekrnann Jörg H
Published: November 2023

Most knowledge-based systems in artificial intelligence (AI) with a commitment to a symbolic representation support one important operation: ‘matching of descriptions’. This operation, called unification in the field of deduction systems, is the ‘addition and multiplication’ of many AI systems, and is consequently often supported by special purpose hardware or by a fast instruction set. Unification theory provides the formal framework for investigations into the properties of this operation, which is in essence the solving of equations in an (equationally defined) free algebra.