bluuebunny/crossref_metadata_2025_split huggingface dataset:
A look inside Reduce
Authors: MacCallum Malcolm A H , Wright Francis J 
Published: November 2023

Abstract
To understand what we find when we look inside an algebra system we must first have some idea what the data is that the system is representing. For emphasis, we repeat here that although the systems are performing calculus, they are not doing analysis; that is, they are emphatically not taking limits, testing convergence, and so on, except where those operations can be done by a strictly algebraic method: this is of course just what humans do when performing calculus. As mentioned in the first chapter, all of us will happily differentiate elementary functions without going back to first principles and taking limits, because we know the computational rules, like the rule for differentiating a product, which ensure that the results are correct, and we know that all the usual functions obey these rules. The rules are all in fact algebraic in nature.
Ranked Universe
Authors: Borovik Alexandre , Nesin Ali 
Published: November 2023

Abstract
Properly speaking, the book begins with this chapter. In Section 4.1 we define the notion of a ranked universe. We leave to the Appendix C and [285] the verification that the ranked groups are in fact the groups of finite Morley rank, as the original definition of Morley rank plays no particular role here. The key element of the definition is Axiom A, a noetherianity condition. We also state as Conjecture 4.7 the main conjecture, discussed in the Preface, which has motivated most of the work discussed here. Section 4.2 contains the first consequences of the definition, and for many of them it may be said that if they did not follow from the definition given, we would be forced to consider them as additional axioms. (Section B.5 contains a discussion of some desirable properties that may not follow from the definitions.) Finally Section 4.3 introduces the more technical notion of uniform definability, corresponding to the notion of an algebraic family of algebraic varieties in algebraic geometry. The exercises for Section 4.2 will enable the reader not accustomed to the model theoretic point of view to develop a better appreciation of the content of the formalism presented, and at the same time to check his or her understanding of the notions presented in Chapters 2 and 3. The first substantial results concerning groups of finite Morley rank (ranked groups) are found in the next chapter.
Bad Groups
Authors: Borovik Alexandre , Nesin Ali 
Published: November 2023

Abstract
A remarkable result about algebraic groups asserts that the Borel subgroups (i.e. the maximal connected solvable closed subgroups) of an algebraic group Gover an algebraically closed field are conjugate to each other (see e.g. [189, §21.3]). Moreover, if G0 is nonnilpotent, then its Borel subgroups are also nonnilpotent [189, §21.4]. The situation in groups of finite Morley rank is not so well-understood. We can introduce the notion of a Borel subgroup of a group G of finite Morley rank in almost the same words: B :S G is a Borel subgroup if B is maximal among the connected definable solvable subgroups of G. However the proof of the conjugacy of Borel subgroups in algebraic groups cannot be generalized to the context of groups of finite Morley rank, because the former is based on the notion of complete variety from algebraic geometry, which does not have a model theoretic analogue.
Möbius Transformations and Inversion
Authors: Needham Tristan 
Published: November 2023

Abstract
Thus, although more than 150 years have passed since August Ferdinand Mobius first studied the transformations that now bear his name, it is fair to say that the rich vein of knowledge which he thereby exposed is still far from being exhausted. For this reason, we shall investigate Mobius transformations in considerably greater depth than is customary.Clearly it would be neither appropriate nor feasible for us to explore this connection in detail, but let us at least briefly indicate how Möbius transformations are related to Einstein’s Theory of Relativity.
Further Geometry of Differentiation
Authors: Needham Tristan 
Published: November 2023

Abstract
In the previous chapter we began to investigate the remarkable nature of analytic functions in C by studying mappings in the less structured realm of ℝ. In particular the Jacobian provided us with a painless way of deriving the Cauchy-Riemann characterization of analytic functions, and also of computing their amplitwists. However, this approach was rather indirect. In this chapter we will instead study differentiation directly in the complex plane, primarily through the use of infinitesimal geometry. Our first application of this approach will be the rederivation of the Cauchy-Riemann (henceforth “CR”) equations, and the discovery of new forms that they can take on.
Winding Numbers and Topology
Authors: Needham Tristan 
Published: November 2023

Abstract
In this chapter we shall investigate a simple but immensely powerful concept—the number of times a loop winds around a point. In Chapter 2 we saw that this concept was needed to understand multifunctions, and in the next chapter we will see that it plays an equally crucial role in understanding complex integration. However, only the first two sections [up to (2)] of the present chapter are actually a prerequisite for that work; the rest may be read at any time. If you are in a rush to learn about integration, you may wish to skip the rest of the chapter and return to it later.
How to silence a universal machine
Authors: Smullyan Raymond M 
Published: November 2023

Abstract
As further applications of diagonalization, we now turn to a fascinating variety of problems that generalize some results in recursion theory known as halting problems. That they really do generalize theorems in recursion theory will be seen in a later chapter. The results of this chapter are also related to incompleteness theorems, as will be seen in the next chapter.
Epistemic Aspects of Databases
Authors: Winslett Marianne 
Published: November 2023

Abstract
This chapter explores the potential use of logic, and of the belief revision techniques of the preceding chapter, in databases and knowledge bases. In Section 1, we show how to formulate relational databases in first order logic, and discuss the use of logic in studying database queries, updates, incomplete information, and integrity constraints. Section 2 concentrates on the problem of updates to knowledge bases, a generalization of databases in which integrity constraints and incomplete information may combine to make updating very difficult. Section 3 shows how logic can be applied to the problems of diagnosing faults in circuits and of reasoning about the effects of actions.
Towards a Computational Treatment of Time
Authors: Gabbay D M , Reynolds M 
Published: November 2023

Abstract
This chapter is about topics relevant to the algorithmic handling of time, both in pure logic and in computing. It attempts to present a unified point of view, where computational and mathematical properties of the logic of time are interwoven in an intrinsic way. This is not difficult to do. Time is a very exciting subject. It is one of the few subjects on which everyone is an expert. We move through time continuously and in order to survive and manage ourselves sensibly we constantly have to make temporal decisions. Philosophy, since the days of Aristotle, has been trying to analyse the way we make these decisions. With the rise of computer science, where ideally one wants the machine to do the job for or of the human, there is a new urgency in the precise logical analysis of human temporal activity.
Non-monotonic Temporal Reasoning
Authors: Sandewall Erik , Shoham Yoav 
Published: November 2023

Abstract
Temporal reasoning is not restricted to reasoning about time itself; it also includes reasoning about phenomena that take place in time, i.e. reasoning about actions and change. Such reasoning is required for a number of purposes in Al programs, in particular for prediction, planning and plan modification, plan recognition, and diagnosis. These types of reasoning tasks depend crucially on the use of action laws, i.e. axioms which specify what effects an action will have or may have in a particular situation. Besides the action laws, each reasoning task typically involves (1) designation of certain actions which have been (will be, may be) performed, as well as their order of execution; (2) statements about the state of the world before the actions; (3) statements about the state of the world after the actions. In a planning task (2) and (3) are given and (1) is sought; in a prediction task (1) and (2) are given and (3) is sought, and so on.
Congruence axioms
Authors: Roe John 
Published: November 2023

Abstract
Affine geometry by itself does not capture the full richness of our intuitive picture of geometry. Its chief deficiency is that we are only able to compare lengths along the same line by means of the ruler axiom; and though we can extend this to compare lengths along two parallel lines (by drawing appropriate parallelograms), it makes no sense in affine geometry to compare lengths AB and AC if AB and AC are not parallel.
The Mathematical Study and Approximation of Semi-conductor Models
Authors: Jerome Joseph W 
Published: November 2023

Abstract
One may view classical or semi-classical semi-conductor modelling as a pyramid structure, in which the Boltzmann Transport Equation (BTE) forms the summit and the drift-diffusion system the base; intermediate are systems derived by taking moments of the Boltzmann equation. The hydrodynamic model and its momentum relaxation time limit, called the energy model, are examples. These will be discussed more fully in the final technical section of this chapter.
The rational line
Authors: Huxley M N 
Published: November 2023

Abstract
The rational numbers form a countable set, dense in the real numbers. They are set in the real line like stars in the sky ‘to illuminate the mystery of the continuum’ (a remark attributed to Borel). In any region, if magnified sufficiently, faint stars appear. To explain this bright and faint analogy, we need the concept of height.
Exponential sums with modular form coefficients
Authors: Huxley M N 
Published: November 2023

Abstract
Modular forms, and their generalization to automorphic functions, lie at the centre of mathematics, in the sense that they appear in combinatorics, group theory, number theory, algebraic geometry, geometric topology, functional analysis, and complex analysis. Swinnerton-Dyer once declared that the unsolved problems in mathematics remain unsolved because we have not yet connected them to modular forms.
Exponential sum theorems
Authors: Huxley M N 
Published: November 2023

Abstract
We resume the treatment of the simple exponential sum in one variable. from Chapter 7, obtaining the results of Huxley (1993a), extended to short intervals and congruence families. We use the notation of Chapter 7, and assume all the restrictions made there on the parameters N (length of short sums) and R (expected size of the denominator). We have estimated the numbers A and B in Lemma 7.4.3.
Symmetric Domains Of Tube Type
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
The tube domains considered in the preceding chapter are multidimensional analogues of the upper half-plane in C. In this chapter, for the case of tubes over symmetric cones, we introduce the analogue of the fractional linear transformation which maps the upper half-plane onto the unit disc; in the special case of the matrix domains this map coincides with the classical Cayley transformation. We give a description of the image domain, its kernel function, its Shilov boundary, its Bergman metric, and its group of holomorphic automorphisms.
Conical And Spherical Polynomials
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
As before, in this chapter V will be a simple Euclidean Jordan algebra, Vℂ its complexification; we will asume that the inner product on Vℂ is given by (z/w) = tr(zw). We will denote by 𝒫 the space of all polynomials on Vℂ. When convenient, we will also think of 𝒫 (without changing notation) as the space of complex-valued polynomials on the real vector space V. A natural representation π of the group GL(Vℂ) is defined on 𝒫 by
Taylor And Laurent Series
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
In this chapter we use the results developed in Chapter XI to study series expansions of functions naturally associated with the geometry of a symmetric cone. In the first section we consider the important special case of series representing K-invariant analytic functions in a neighbourhood of the origin. In subsequent sections we define Laurent expansions for functions analytic in U-invariant domains in Vℂ and we extend the theory of expansions to include functions that are not necessarily K-invariant. We also describe the hull of holomorphy of U-invariant domains.
Invariant Differential Operators And Spherical Functions
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
In Section 1 we describe the linear differential operators on Ω which commute with the action of G. We give explicit systems of generators for the algebra 𝔻 (Ω) of these operators. We also discuss the effect of the map x ↦ x−1 on them. In Section 2 we exploit the fact that each element of 𝔻 (Ω) preserves each K-irreducible subspace 𝒫m of the polynomials on V, and acts on it as a scalar; we compute this scalar for a system of generators of ID&gt;(O). As an application we obtain further characterizations of the Hilbert spaces ℋν (D) of holomorphic functions introduced in Chapter XIII, including explicit formulas for their norms. In Section 3 we determine the spherical functions on Ω, i.e. the K-invariant joint eigenfunctions of the invariant operators. In Section 4 we define the spherical Fourier transform on Ω and compute some examples. In Section 5 we compute the Harish-Chandra c-function for Ω and we state the spherical Fourier inversion formula without proof.
Special Functions
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
It is strongly suggested by the generalized binomial expansion of Chapter XII that a multivariable version of the hypergeometric function can be associated with every symmetric cone. The theory of these functions is the subject of the present chapter. After proving some general results we look in more detail at the special cases of generalized Bessel functions and Gauss hypergeometric functions. We establish some connections with spherical functions on the tube TΩ and we prove the generalization of a number of classical results including a rather complete theory of the Hankel transform and of Laguerre polynomials.
Wave statistics and spectral method
Authors: Rahman Matiur 
Published: November 2023

Abstract
In the previous chapters we have discussed the waves that are regular and remain invariant in their properties from one cycle to the next. However, ocean waves are in general not regular but random in the sense that the ocean surface is composed of waves moving in different directions and with different amplitudes, frequencies and phases. In this situation ocean waves should be described by their statistical properties. In the following section, therefore, we shall define the statistical parameters from a mathe matical viewpoint. These statistical definitions are then applied to the random wave field.
Functional Analysis And Wavelets
Authors: Holschneider M 
Published: November 2023

Abstract
In the first section we saw that the space of C∞ regular, localized functions with all moments vanishing is characterized by a rapid decay of the modulus of the wayelet coefficients at small and large scales and at large positions so that the wavelet transform is localized over the position-scale half-plane. By duality the space of tempered distributions was characterized by an at most polynomial growth of the wavelet coefficients at small and large scales and large positions. A third space, . the Hilbert space of square integrable functions, was c aracterized by a square integrable distribution of the energy over the position-scale half-plane. In all three cases this characterization was independent of the analysing wavelet. It therefore seems natural to introduce spaces that are characterized by their distribution of the modulus of the wavelet coefficients over the position-scale half-plane. This will be done in this chapter. For supplementary information we refer the reader to the literature. In particular in the case of the wavelet transform the books by Meyer (1985) and Coifman and Meyer are certainly standard references. On the other hand, many ideas that are now unified under the wavelet approach can be found in great detail in the book by Triebel (1992). Again, all results can be generalized in the language of group theory, as was shown by Feichtinger and Grochenig (1989).
Applications to compositions of polynomials
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
As the main matter of this part of the book, we have considered a polynomial f and investigated the location of the zeros of f′ relative to those of f . The methods which we have established for this purpose, in particular, those of Chapters 3 and 4, are of a much wider scope. Instead of f′, we may consider a polynomial h which is ‘derived’ from for several polynomials f1,.. .,fk by a certain operation.
Polynomials with real zeros
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
Definition 6.1.1 Let f be a polynomial of positive degree with only 1wl zeros. Then the length b a of the smallest internal I(f) := [a, b] containing all the zeros of f is called the span of f. The span is zero if and only if the zeros of f are coincident. If f is of degree n ≥ 2, then, by the Gauss-Lucas Theorem 2.1.1, I(f߰) ⊆ I(f). In other words, if Dn and Dn™1 denote the spans of f and f’, respectively, then Dn-1 Dn.
Inclusion of some of the zeros
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
Some of the previous results may be interpreted as estimates in terms of a norm. For a polynomial the L2 norm on the unit circle is given by
Using the estimate (8.1.2) of Theorem 8.1.7 with A = 1 and p = 2, we find that, for any zero z* of f, we have
(9.1.2)
Number of zeros in a domain
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
For counting the zeros of a real polynomial f in an interval, one important approach was to study the behaviour of f(x) and that of certain auxiliary polynomials as x passes through a zero (see the theorems of Budan Fourier and Sturm and their proofs). The situation is quite different when we want to count the zeros of a complex polynomial f inside a simply connected domain Ω We do not know a path that passes through the zeros. However, since f has only a finite number of zeros, we may always replace Ω by an appropriate bounded domain Ω′ whose boundary is a Jordan curve, γ that surrounds the zeros to be counted. This suggests that we employ the principle of the argument (Theorem 1.1.3). It remains to study how the argument of f(z) changes as z traverses γ. For this purpose, we shall employ the concept of a Cauchy index. We start with a heuristic consideration.
Growth estimates
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
In this chapter, we shall discuss how fast a polynomial f can possibly grow in the complex plane. Assuming that f is of degree n, we need to know something about f(z), such as its modulus, its real part, or its imaginary part on a setε, containing at least n + 1 points. Except for this requirement, the set ε, may be almost any bounded subset of ℂ. We wish to find out how large its modulus or its real part (or its imaginary part) can be at a given point outside £,, or on any other set. For example, we may suppose | f (z) | to be bounded above by M on the unit circle and look for its sharp upper bound on the concentric circle of radius R &gt; 1.
We start with the Bernstein Walsh lemma, which provides us with a method for handling very general situations. We shall present two other methods, namely, 1The Convolution Method’ and ‘The Method of Functionals’. The latter two yield more precise results, but they are not as general as the first method.
Mean values
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
In this chapter, we study the growth of a polynomial in terms of a variety of mean values. We mention a few examples in order to illustrate the kind of problems we shall discuss. Suppose that we know the values of a polynomial f, of degree at most n, at n + 1 distinct points z0, ... , z11, and that we are given a point ζ other than all the zv How large can |f (ζ) | be in comparison to
Derivative estimates on the unit disc
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
Let there be given a polynomial of degree at most n and some meaningful information about the kind of values it takes on a prescribed subset ε of the complex plane. What can we say about the size of its derivative? It will soon become clear that this question is closely related to the one considered in the two preceding chapters. The special case of the problem where ε is the unit circle is particularly interesting, and has received a good deal of attention. This is partly because it leads us in a natural way to the consideration of a similar question about trigonometric polynomials and entire functions of exponential type.
Automorphisms
Authors: Pasini Antonio 
Published: November 2023

Abstract
An automorphism of a geometry r is an isomorphism from Γ to Γ. The automorphisms of Γ form a group. We denote this group by Aut(f) and we call it the full automorphism group of Γ, also the automorphism group of Γ, for short, using the definite article ‘the’. When we speak of some (or any, every ...) subgroup of Aut(f), then we use the indefinite article ‘a’ (or the words ‘some’, ‘any’, ‘every’ ...).
Coxeter Diagrams
Authors: Pasini Antonio 
Published: November 2023

Abstract
We are now ready to come back to the main theme of Chapter 7: recovering geometries from diagrams. We are not going to offer an exhaustive account of all theorems where some description is achieved for some class of geometries with a given diagram; this would be too long. We only want to show how the tools prepared in chapters 8-12 can be used to continue the work started in Chapter 7, with no aim to finish it.
Constructing finite p-groups
Authors: Leedharn-Green C R , McKay S 
Published: November 2023

Abstract
This chapter begins the process of constructing p-groups of coclass r as given by Leedham-Green (1994b). A class of ߢconstructibleߣ p-groups is defined, and the structure theorem will say that every p-group P of coclass r has a normal subgroup N, of order bounded in terms of p and r, such that P/N is constructible. Recall that it has already been shown in Chapter 6 that a p-group of coclass r has a subgroup of nilpotency class 2 and small index. A constructible p-group is obtained from a finite homomorphic image of a p-adic space group of finite coclass by Twisting' the image of the translation subgroup, thus making it of nilpotency class 2. Section 8.1 presents the basic theory involved in carrying out this ‘twisting’ operation in the context of central extensions of modules for a group. The structure theorem will be proved in Chapter 11 after p-adic space groups have been investigated in some detail in Chapter 10.
Homological algebra
Authors: Leedharn-Green C R , McKay S 
Published: November 2023

Abstract
Homological algebra for our purposes consists of studying certain abelian groups Πn (G, A) and Hn (G, A), known as the nth cohomology and homology groups of the group G with coefficients in the G-module A. Of these, cohomology is the more important. We shall not need the nth cohomology groups for n greater than :3, but it is convenient to develop the theory in general covering all n ∴ 0, and to consider derived functors in general.