bluuebunny/crossref_metadata_2025 huggingface dataset:
Invariant Differential Operators And Spherical Functions
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

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>(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

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

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

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).
Hall-Littlewood Symmetric Functions
Authors: Macdonald I G
Published: November 2023

Let x,...,x and t be independent indeterminates over Z, and let A be a partition of length < n.
Applications to compositions of polynomials
Authors: Rahman Q I , Schmeisser G
Published: November 2023

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

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

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

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

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 > 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

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

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.
Basic Concepts and Examples
Authors: Pasini Antonio
Published: November 2023

In particular, a rank 2 geometry is just a connected bipartite graph where every vertex is adjacent to at least two vertices. A geometry of rank 3 is a connected 3-partite graph where the neighbourhood of every vertex is connected and every edge is contained in at least 2 cliques of size 3.
More Examples
Authors: Pasini Antonio
Published: November 2023

We first describe some ‘small’ rank 3 geometries. We say they are small not because they are ‘globally’ small (some of them in fact have infinitely many elements), but because they ‘locally’ look small: the residues of their elements are ordinary polygons or ordinary digons.
Automorphisms
Authors: Pasini Antonio
Published: November 2023

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

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

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

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.
The structure of finite p-groups
Authors: Leedharn-Green The structure of finite p-groups [CT] C R , McKay S
Published: November 2023

The general aim of this chapter is to prove that if P is a finite p-group of coclass r then P has a normal subgroup N of order bounded by a function of P and r such that P/N is constructible as defined in Definition 8.4.9. This generalizes the theory of P-groups of maximal class described in Section 8.2; in particular, it generalizes the observation that if Pis a 2-group of maximal class and order at leru;t 16 then P ha.-, a normal subgroup N of order 2 such that P/N is dihedral. We give generic bounds in terms of P and r to the order of N and the derived length and other invariants of P.
Beyond coclass
Authors: Leedharn-Green C R , McKay S
Published: November 2023

The primary step in classifying the finite p-groups of coclass at most p lies in the classification of infinite pro-p-groups of coclass at most r. The question then arises as to what other classes of pro-p-groups can be used for finding other classifications of finite p-groups. At this point the pro-p-groups become at least as interesting as the finite p-groups. The conditions to be imposed on the pro p-groups are generally finitistic, and hence apply vacuously to finite p-groups.
Four-Manifolds
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

This chapter falls into three parts. In the first we review some standard facts about the geometry and topology of four-manifolds. In the second we discuss a number of results which date back to the 1960’s and before; in particular we give an account of a theorem of Wall which accurately portrays the limited success, in four dimensions, of the techniques which were being used to such good effect at that time in the study of high-dimensional manifolds. This discussion sets the scene for the new developments which we will describe in the rest of this book. In the third section we summarize some of the main results on the differential topology of four-manifolds which have sprung from these developments. The proofs of these results are given in Chapters 8, 9, and 10. The intervening chapters work, with many digressions, through the background material required for these proofs.
Topology And Connections
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

In Section 5.3 we discuss a different route by which the topologies of X and ℛ* are related: this is through the notion of a ‘concentrated’ or ‘particle-like’ connection, whose curvature is concentrated near a finite collection of points. We have seen in Section 4.3 that such connections can be expected to arise near the boundary of the moduli space of ASD connections. Here we shall examine the topological content of this phenomenon, and its relationship to the Poincare duality pairing on X.
The Differential Topology Of Algebraic Surfaces
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

Let us now review our position with regard to the overall aims of this book. In Chapter 1 we set up the twin problems in four-manifold theory: deciding the existence and uniqueness of manifolds with given intersection forms. In Chapter 8 we have shown how arguments with Yang-Mills moduli spaces give strong non-existence results, and in Chapter 9 we have developed invariants of four–manifolds aimed at the complementary question of uniqueness. However we have not yet given any full-blooded example of the application of these invariants, i.e. shown that they can distinguish differentiable four-manifolds with the same intersection form, and this is the purpose of the present chapter. We will describe the main ideas in the proof of the general Theorem (1.3.5), and give a complete proof of the special result ( I .3.6).
Division systems and division spaces
Authors: Henstock Ralph
Published: November 2023

Generalized Riemann and variational integration theory connect a base space T of points of some kind, a space K of values, and three structures, (a) division systems and division spaces in T, (b) the algebra of K, (c) a topology or an order in K to define limits. As (a) is independent of (b) and (c) together, we can consider many combinations of the first structure with the other two.
Limits Under The Integral Sign, Functions Depending On A Parameter
Authors: Henstock Ralph
Published: November 2023

It is well known that the Lebesgue integral handles with ease many problems of the commutability of operations of integration and the taking of a limit, partially resolving C. Jordan’s question quoted by de la Vallee Poussin (1892a).

Give a rigorous theory of differentiation under the integral sign of definite integrals, with precise conditions which limit Leibnitz’s rule, principally for unbounded regions of integration or unbounded functions, and particularly many celebrated definite integrals.
Differentiation
Authors: Henstock Ralph
Published: November 2023

In Section 3.1 one of the limit processes is that of differentiation by a parameter, of an integral over a fixed elementary set E, (3.1.2) giving the one dimensional result. We can obtain necessary and sufficient conditions for

(3.1.2) to hold by translating into the present mathematical language the general Theorems 3.1.3 and 3.1.4.
Cartesian Products Of A Finite Number Of Division Systems (Spaces)
Authors: Henstock Ralph
Published: November 2023

Theorem 5.1.2 appeared first in Henstock (1961b), and the proof has been refined greatly since then. See also Kurzweil (1973a). Theorem 5.1.3 is due to T. W. Lee (1971/1972). In Lebesgue integration Theorem 5.1.2 in the symmetrical form is essentially due to Fubini (1907), and the unsymmetrical form is due to Cameron and Martin (1941) and Robbins (1948). See also Bosanquet (1930a).
Excesses and Coverings
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

There is a natural duality between packing triples on a set so that every pair appears at most λ times, and covering the pairs by triples so that every pair appears in at least λ triple. A covering by triplesof order vand index λ or CT( v, A),is a set V of velements, and a collection (multiset) of 3-elcment subsets of V(triplesas usual) for which every unordered pair of elements appears in at least Aof the triples.
Support Sizes
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Triple systems are permitted to have repeated triples, triples that arc identical as 3-subsets. A theme throughout studies of triple systems is the goal to produce simpletriple systems having no repeated blocks. However, in some applications, the presence of repeated triples poses no difficulty. In fact, for certain applications in experimental design (Wynn, 1977; Foody and Hedayat, 1977), it is desirable to have many triples repeated.
Chromatic Number
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Erdos and Hajnal (1966) and Lovász (1968, 1968a) were apparently the first to consider (weak) vertex-colourings of hypergraphs. Somewhat later, Berge (1973) formalized the notions of weak and strong chromatic numbers of hypergraphs. Both arc natural generalizations of the chromatic number of a graph: view a proper vertex-colouring of a graph either as one in which no edge is monochromatic (weak colouring)or as one in which the vertices incident upon a given edge all receive different colours (strong colouring).
Semigroup diagrams and word problems
Authors: Higgins Peter M
Published: November 2023

One of the few outstanding problems in algebraic semigroup theory that can be explained to a layman in a few minutes is the one-relator word problem. You are given two words, w1 andw2, together with another pair of words,u andv, all of whose letters come from some alphabetX. Is it possible to transform the word w1 into the wordw2, given thatu andv are treated as ‘equal’, meaning thatu andv can be interchanged, one for the other, as often as you please? The answer will of course depend on the given words. For example, let w1=ab2, w2 =ab2a2, with our ‘relation’ given byab= aba.
Around the spectrum
Authors: Helemskii A YA
Published: November 2023

The Latin word spectrum means something like “apparition”, “spirit” or “ghost”. But whereas the ghosts of kings and dukes, as a rule, inhabit ancient half-abandoned castles, the habitat of operator ghosts and, more generally, that of the elements of abstract algebras, is the complex plane, to which they were confined by Hilbert in his classic definition. In both cases the ghosts, if asked properly, can tell us many interesting things. Hamlet, the Prince of Denmark, in his day found this out in conversations with his father’s ghost. We hope that our reader will be no less impressed by the theorems on the representation of skew fields with continuous inverse (Theorem 1.17) and on the holomorphic functional calculus of a Banach algebra element (Theorem 2.15).
Polynormed algebras that are near to Banach algebras
Authors: Helemskii A YA
Published: November 2023

Historically, mathematicians became interested in general polynormed algebras (= locally convex algebras), as opposed to just Banach algebras, essentially for the same reasons which caused the theory of locally convex spaces to appear at a certain stage in the development of the theory of Banach spaces. The basic reason was, evidently, connected with applications: there had been an accumulation of significant “non-Banach” examples.

Topologists needed to work with the algebras C(Ω) when Ω was non-compact, complex analysists needed to work with algebras of holomorphic functions defined on domains in ℂn (specialists in operator theory needed to work with problems in functional calculus that also involved these same algebras), specialists in differential equations and harmonic analysis needed to work with various algebras of test functions and distributions. Another, this time intrinsic, stimulus was that studying structures that are more general than Banach algebras enables one to get a better understanding of the results of the “Banach” theory itself and to establish natural limitations to the scope of possible generalizations.
The polynomial 3-primes problem: an asymptotic solution
Authors: Effinger Gove W , Hayes David R
Published: November 2023

We begin our analysis of (6.3) with six preliminary results which will set the stage for the approximations in the next section. Our first lemma generalizes the righthand inequality of (1.23).
Introduction
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

Here we collect the chapter headings in the rest of the book and give a brief summary of the contents of each chapter.1 Historical overview A very brief survey of the subject of summability of series, ranging from the earliest attempts to sum convergent infinite series up to the first attempts to formalize the theory of summability of divergent series. For some of the material here we rely on a set of unpublished lecture notes of L. S. Bosanquet T[5].
Historical overview
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

Infinite series, whether used explicitly or implicitly, have fascinated mathematicians for centuries. In fact, convergence was familiar to Archimedes as early as 250 - 200 BC.
Introduction
Authors: Yves Diers
Published: November 2023

The motivation for a study of the properties of categories of commutative algebras was the natural outcome of the publication of several papers developing techniques which mimic those of classical commutative algebral and algebraic geometry in areas such as the study of ordered algebra, lattice algebra, real algebra, differential algebra, and C∞ -algebra. For instance, G. W. Brumfiel’s Partially ordered rings and semi-algebraic geometry [4] develops a kind of ordered commutative algebra, the paper by K. Keimel entitled ‘The representation of lattice ordered groups and rings by sections in sheaves’ [28] develops a kind of lattice-ordered commutative algebra, the paper by M. E. Alonso and M. F. Roy entitled ‘Real strict localizations’ [1] develops real commutative algebra, the paper by W. K. Keigher entitled ‘Prime differential ideals in differential rings’ [27] develops differential commutative algebra and the paper by I. Moerdijk and G. E. Reyes ‘Rings of smooth functions and their localizations’ [32] develops C∞ commutative algebra. There are many other similar papers intent on extending the domain of interpretation of commutative algebra and transcending the particularity of ring structure. Thus the idea arose that the essence of commutative algebra is to be sought not in its calculus in commutative rings, but rather in the universal calculus in the category of commutative rings. It became natural to look for the universal properties in categories of commutative algebras which make commutative algebra work.
Schemes
Authors: Yves Diers
Published: November 2023

The localization process associates to any object a sheaf of local objects on its prime spectrum, while the globalization process rebuilds any object from the continuous family of its localized objects. By means of these processes, any object turns into a geometrical object: its affine scheme. By making these processes functorial, we obtain an equivalence between the category of affine schemes on a Zariski category A and the dual of A. Numerous properties of objects and morphisms in A have nice geometrical interpretations in the associated category of affine schemes. For example, a morphism in A is a regular epimorphism (or singular epimorphism etc.) if and only if its affine spectrum is a closed immersion (or open immersion etc.). Schemes on a Zariski category are built up by the classical method of gluing affine schemes together using modelled spaces and locally modelled spaces instead of ringed spaces and locally ringed spaces. Numerous classical properties and constructions of schemes are valid, finite limits and arbitrary coproducts of schemes exist, and the general gluing construction can be performed. Moreover, separated schemes are disjunctable; hence all disjunctors exist in the category of separated schemes and they are open immersions. In particular, affine schemes are disjunctable, and the open immersions into an affine scheme are precisely its singular subobjects.
Various notions of stability
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

In this chapter we discuss the interrelations between various notions of stability, both those that have already appeared in the literature and also new ones suggested by our attempts at characterization.
Introduction
Authors: Werschulz Arthur G
Published: November 2023

Most problems of scientific computation must be solved with uncertainty. Prime examples include the solution of differential equations (DE) and integral equations (IE). We typically have only partial and contaminated information about such problems, this information being priced. We motivate these ideas by considering the solution of the Poisson equation.
Example: a two-point boundary value problem
Authors: Werschulz Arthur G
Published: November 2023

We use a simple example of a two-point boundary value problem (Lu)(x) = -u"(x) + u(x) = f(x)  u'(O) = u'(l) = 0. for O < X < 1, to illustrate major issues and concepts of information-based complexity. First, we must describe our problem more carefully. That is, we specify the problem formulation, permissible information, and a model of computation.
General formulation
Authors: Werschulz Arthur G
Published: November 2023

Our main goal in this book is to find approximate solutions of differential and integral equations at minimal cost. Such approximations are given by optimal complexity algorithms. Instead of starting from scratch when finding optimal complexity algorithms for each new class of problems, it is more efficient to first recall general results from the theory of information based complexity, which we can then apply to each class of problems. As in the previous chapter, our key concepts are the problem formulation, the permissible information that may be used to solve the problem, and the model of computation that tells us which operations are permissible, and at what cost. We also introduce several settings, which allow us to measure the error and cost of algorithms. In particular, we are interested in the worst case setting (error and cost are measured at a hardest problem element), the average case setting (error and cost are defined on the average), and the probabilistic setting (error and cost in the worst case, but disregarding a set of problem elements of given measure).
Complexity in the asymptotic and randomized settings
Authors: Werschulz Arthur G
Published: November 2023

Recall that our emphasis so far has been on finding an algorithm that computes an ε-approximation over a set of problem elements, this computation being done at minimal cost. Moreover, we have only considered information and algorithms that are deterministic, not allowing for randomness. We now look at what happens if we measure what happens at a fixed problem element as the algorithms get successively better (the asymptotic setting) or if we allow nondeterministic algorithms (the randomized setting). We first look at the asymptotic setting. In this setting, we require that a sequence of algorithms approximate the solution at a particular problem element, and that this sequence converge as quickly as possible. The main result is that algorithms and information that are optimal for linear problems in the worst case setting are also optimal in the asymptotic setting.
Semigroup Kernels
Authors: Robinson Derek W
Published: November 2023

This chapter concentrates on general properties of the kernels, properties which are valid for all the kernels constructed from strongly elliptic operators. Then, in Chapter IV, we specialize to second-order operators with real coefficients. The semigroups generated by this latter class of operators have numerous special properties of positivity and dissipativity which can be exploited to refine the general properties.
Second-Order Operators
Authors: Robinson Derek W
Published: November 2023

The previous three chapters were devoted to the general theory of strongly elliptic operators but this chapter is much more specific. Its aim is the analysis of second-order operators with real coefficients, and the main em phasis is on unimodular groups. The previous theory of course applied to second-order operators but they also have many characteristic features not shared by their higher-order counterparts. For example, Theorem 111.5.1 established that the semigroup kernel corresponding to a strongly elliptic operator is positive if, and only if, the operator is of second-order with real coefficients. This positivity is just one aspect of the special properties of the second-order operators. Dissipativity is another.
A remark on the derivation of flocks
Authors: Lunardon G
Published: November 2023

Applying the derivation to the known examples of flocks, it has been proved that in many cases the derived flocks are isomorphic to the given one (see Bader to appear; Bader et al.
On a problem of Wielandt and a question by Dembowski
Authors: Siemens J , Webb B
Published: November 2023

In Dembowski’s book (page 212) a similar question appears in a more restricted situation: If G is a group of automorphisms of a finite projective plane, acting primitively on lines, is G then necessarily primitive on points? The answer to this question is now known, due to the classification of projective planes with primitive collineation group by Kantor (1987).
The Gray Code Function
Authors: Clarke Francis
Published: November 2023

The Gray code of a positive integer m can be defined in terms of its binary expansion as follows. Shift m to the right, discarding the last digit, and then add the result to m without carrying (an exclusive or). Thus if we write ® for the exclusive or operation and ÷ for integer division, so that m÷2 is the right shift of m, then the Gray code of m is g(m) = m ® (m÷2). For each k, the function g permutes the integers between 0 and 2 k – 1. It has the property that the binary expansions of g(m) and (g(m+1) always differ in precisely one digit. This is true cyclically, in the sense that it holds for g(2 k – 1) = 2 k-1 and g(0) = 0. This provides a Hamiltonian circuit on the k-dimensional cube and is the basis of the construction of the diagram in Figure 1.
Out of the p-Adic into the Real
Authors: Everest G R
Published: November 2023

The theorem of this paper, when applied to the case of a profinite group, is well known, although a proof is not easily located. As a basic result on such groups, we feel that it deserves an explicit statement in the literature. We also give some applications to the continuous representation theory of a profinite group. We would like to thank the numerous people who made helpful comments on this result.
Packages for special occasions
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

Having given a detailed (but not comprehensive) overview of the SHEEP and CLASS! commands, we can now get down to applying these programs to specific problems in general relativity. The first problem we shall consider highlights the use of the MATTER package, written by Egils sviestins at Stockholm University.
Anholonomic frames
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

For most calculations in general relativity it is normally better to use anholonomic frames (tetrads), especially frames with respect to which the metric components are constants. Orthonormal frames are of particular importance since they represent the local frame of an observer whose four-velocity is the time like vector of the frame. Null frames also play a significant role and, to a lesser extent, half-null frames. Apart from their physical relevance, the components of the various tensors on these frames will usually be simpler and easier to handle than the corresponding ones with respect to the coordinate frames, since the “coordinate junk” has been eliminated.
Computation of the connection and the curvature
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

One of the earliest general relativity calculations performed with the aid of computer algebra was the computation of the Christoffel symbols and the components of the curvature tensor, Weyl tensor, Ricci tensor and Einstein tensor from the components of a given covariant metric tensor 9ij in some coordinate system. These elementary calculations provide a good illustration of the importance of the choice of algorithms for computer algebra programs in general relativity, a consideration that often seems to have been overlooked. Programs for performing the above calculations were usually based on standard formulas given in general relativity or tensor calculus textbooks. An examination of these formulas reveals that there is a variety of ways of computing the curvature tensor for example.
Computer algebra aided integration of field equations with the NP package
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

The usefulness of the Newman-Penrose (1962) (NP) formalism has been well-proven since its introduction, especially in the field of exact solution theory (see for example Kramer et al. 1981). In many instances, the manipulation of a system of first order differential equations in twenty-four complex indeterminates (i.e. the so-called spin coefficients and curvature quantities) is more conceptually straightforward than working in standard tensorial notation. However, due to the loss of compactness, expressions are often produced which are far too large to be dealt with effectively in hand calculations; this is often the case even when the desired result is simple and compact.
Introduction to reliability
Authors: Ansell J I , Phillips M J
Published: November 2023

The objective of this book is to describe elements of statistical data analysis which are useful in reliability studies, in a manner which gives the flavour of modern approaches. The chapter commences with a description of three examples of reliability studies taken from the authors’ practical experience of statistical data analysis. These studies, which cover a range of industrial contexts, are used later in Chapter 10 to illustrate the concepts presented throughout the book. It is important to define the objectives of a study and the objectives that can be achieved through the application of probability and statistical methods are outlined. The different forms of the data. encountered to achieve the objectives of a reliability study are illustrated in Section 1.4. These illustrations are made using examples from recently published papers in reliability journals and include right censored data, truncated data, accelerated failure data, and data from repairable systems. All these data must be collected and the proper planning of the way the, ae are collected is discussed. This leads to the consideration of databases, both the use of historic databases and the construction of new ones. However, before commencing any discussion of statistical methods a formal definition of reliability is required. The reliability of a system (or component) is defined as the probability that the system operates (performs a function under stated conditions) for a stated period of time. Usually the period of time is the initial interval of length t, which is denoted by [O, t)
Lifetime distributions
Authors: Ansell J I , Phillips M J
Published: November 2023

This chapter introduces the basic ideas of parametric lifetime distributions, by firstly defining expectation, the median, and the mode, and then giving details of the most important examples encountered in reliability. These are the gamma, Weibull, extreme value, and normal distributions. In all cases the probability density functions are illustrated and for the gamma and Weibull cases the hazard functions are illustrated. After a treatment of the various kinds of censoring mechanisms that can produce examples of censored data, the problem of estimation is considered from the non-parametric standpoint. The well-known Kaplan-Meier and Nelson-Altschuler estimators for right censored data are described. When using parametric lifetime distributions values of the parameters have to be chosen. The ‘best’ values have to be chosen (estimated) on the basis of the information from the data available. The various methods of estimating parameters in parametric models from the classical, likelihood, and Bayesian viewpoints are explained. The chapter is concluded with an account of hypothesis testing which is applied to the question of the goodness-of-fit of a hypothesized lifetime distribution for censored and truncated samples.
System reliability
Authors: Ansell J I , Phillips M J
Published: November 2023

This chapter introduces the basic ideas for analysing system reliability. In Chapters 2 and 3 methods of analysing single components or systems were presented and no account was taken of the construction of the systems from components (or subsystems). Often there is information on the per formance of these components (or subsystems) and it is sensible to use this information in deciding on the reliability of the system. If data are available on the performance of components it is possible to calculate (estimate) the reliability of these components. Then it is necessary to have methods for calculating (estimating) the reliability of the system constructed from these components. The basic ideas and methods are presented in this chapter. Firstly, the composition (structure) of the system must be defined in terms of its components. This is done using the structure function with the associated ideas of paths and cuts. These are illustrated using simple examples of series and parallel systems, as well as the important k-out-of-n system. The structure function can be used for probability calculations to obtain the reliability of the system. (The reliability function has been introduced as a function of time, t. In this chapter it will be assumed that a particular value of t is of interest, and this will not be changed during the calculations. Hence, for this reason, the time notation will not be used in this chapter.) For simplicity the probabilistic results are restricted to systems with coherent structure functions, which is all that is needed for most practical applications. If the components operate independently and have identical probabilistic mechanisms then the calculation of the reliability of the system is simplified and a number of results follow. One of the most important is that the reliability function of the system of n components is a linear combination of the reliability function of k-out-of-n systems. As these calculations are by no means as easy for non-identical components there is a real need for good methods which can be used to give bounds on the system reliability. The important topic of three-state reliability, which covers the problem of two kinds of failure, is expounded.
Models for repairable systems
Authors: Ansell J I , Phillips M J
Published: November 2023

Previous chapters have dealt with the distribution of lifetimes of single components and the structure of systems. In this chapter the modelling and assessment of systems whose components may be repaired is explored. These are probably the most common type of system studied in reliability engineering. The introduction of repair brings with it a whole realm of new problems. The simplest case is if repair is instantaneous. The system would then be continuously working though it would experience events (failures). The frequency of the failures would be of interest. However, in many cases the repairs will take time and may consist of a sequence of events such as identifying the failure, sending for a repair team, the repair of a component, and the recommissioning of it. Whilst it would be possible to devote the whole chapter to characterizing repair and the types of strategies used, only a brief outline will be given at the beginning of the chapter. At the end of the chapter optimal replacement strategies will be considered. The subject can be studied in more depth by consulting Aven and Bergman (1986), Barlow and Hunter (1960b), and G.L. Glasser (1967). The main focus, though, of this chapter will be on the description of systems behaviour through stochastic processes. Simple stochastic process models which describe the system’s performance will be presented first before more complex models, which are closer to the real world, are considered. These include the modified renewal process, the alternating renewal process, the non-homogeneous Poisson process and the superimposed renewal process. Markovian models are also widely used to model many reliability systems and the analysis of these models is illustrated by methods which use differential equations. The practical application of these models is far from straightforward. There is the need to select the appropriate model and then follow the practicalities of fitting such a model.
Dependency analysis
Authors: Ansell J I , Phillips M J
Published: November 2023

So far it has been assumed that the lifetimes of components (or subsystems) are independent. This chapter explores approaches which do not make this assumption. A recent review of the area has been carried out by Ansell and Walls (1990, 1992). Modelling and the analysis of data from systems where components (subsystems) are dependent are becoming increasingly important as it is realized that dependency may significantly affect the reliability of the system. This has been particularly noted for safety systems when redundancy did not produce the expected level of performance owing to dependency amongst the components. The assumption of dependency leads to a number of problems including its definition. In statistical terms two events are independent if their joint probability is the product of their marginal probabilities so that for two events A and B
Exotic values of G-functions
Authors: Beukers F
Published: November 2023

It turns out that with the conditions imposed so far one cannot say very much in general. However, there are two special cases, both initiated by C. L. Siegel, to which transcendence methods can be applied. The first case is the case of E-functions.
The Tate Conjecture for Almost Ordinary Abelian Varieties over Finite Fields
Authors: Lenstra Hendrik W , Zarhin Yuri G
Published: November 2023

This conjecture has been proved in certain cases, e.g., Fermat hypersurfaces satisfying certain numerical conditions [Sh, Y], elliptic K3 surfaces [AS], 1{3 surfaces of finite height [N, NO] and powers of ordinary K3 surfaces [Z3].
Arithmetic of Subvarieties of Abelian and Semiabelian Varieties
Authors: Vojta Paul
Published: November 2023

We will discuss work of Faltings concerning rational points on subvarieties of abelian varieties. In the talk given at the conference, the present author announced a generalization of this theorem to the case of integral points on semiabelian varieties. The proof has some serious flaws, however, so the generalization still remains a conjecture.
Construction of S4 -fields and modular forms of weight 1
Authors: Crespo Teresa
Published: November 2023

In this paper, we build Galois extensions | Q with Galois group 84, by solving explicitly Galois embedding problems. This construction is applied to the computation of the coefficients of the modular form of weight 1, arising from the representation p, associated to | Q.
Shimura Reciprocity for Modular Functions with Rational Fourier Coefficients
Authors: Laing Alan J
Published: November 2023

We give an exposition of the computational aspects of Shimura’s explicit reciprocity law. First we discuss the classical formulation as in Shimura [4), Thm 6.31, and Lang [2], 11 §1 Thm 1. Then we consider Robert’s formulation in the case that we have rational Fourier coefficients. Finally, we interpret Robert’s formulation in terms of matrices and give an example.
On Serre’s conjecture (3.2.4?) and vertical Weil curves
Authors: Lario Joan-C
Published: November 2023

The first purpose of this contribution is to obtain a general criterion in order to verify this conjecture. In a joint work with P.Bayer [2], we verify (3.2.4?) for the Galois representations defined by the ρ-torsion points of ρ-vertical Weil curves (i.e., modular elliptic curves having bad but potentially good ordinary reduction at ρ, for ρ > 7. Here, we obtain another proof for this case as an application of our general criterion.
A note on Thue’s inequality with few coefficients
Authors: Mueller Julia
Published: November 2023

A proof of this conjecture or even a weaker version such as the number of primitive solutions of (1.1) depends on r or s is out of reach at the present time. What we are able to prove is the following.
Une Relation Ent re les Points Entiers sur une Courbe Algébrique et les Points Rationnels de la Jacobienne
Authors: Hirata-Kohno Noriko
Published: November 2023

Nous donnons ici des minorations de la distance entre des points rationnels distincts d’une variété abélienne. Ces minorations impliquent une majoration de la hauteur des points rationnels de dénominateurs homes sur une courbe algebrique de genre 1 (cf. le théoréme 3.1). Nos résultats donnent alors une version quantitative du theoreme de Siegel sur la finitude des points entiers de telle courbe.
The mixed limit theorem in MPT
Authors: Grenander Ulf
Published: November 2023

A recurrent theme in Part VI will be that pattern inference should be preceded by pattern synthesis: powerful inference algorithms can be constructed, at least in principle, if our understanding and mathematical penetration of the synthesis problem is deep enough. The qualification ‘in principle’ is needed since it may happen that the theoretical optimal algorithms are not feasible with available computer technology.
Symplectic Geometry
Authors: Woodhouse N M J
Published: November 2023

Finite-dimensional symplectic manifolds arise as the phase spaces of classical systems with a finite number of degrees of freedom, where the sym plectic form w is the geometric structure that underlies the definitions of Hamiltonian vector fields and Poisson brackets. Infinite-dimensional sym plectic manifolds arise in field theory. They are less straightforward objects and their study is complicated by some subtle analytical questions. There are, for example, a number of different forms of the no degeneracy condition on w, and for infinite-dimensional manifolds they are not equivalent. For the moment we shall concentrate on the finite-dimensional case.