bluuebunny/crossref_metadata_2025 huggingface dataset:
p-groups of maximal class
Authors: Leedharn-Green C R , McKay S
Published: November 2023

A group of order pn and nilpotency class n − 1 that is, of coclass 1, is said to be of maximal class if n > 4. The basic material about these groups is to be found in (Blackburn 1958)1 and is also covered in the chapter on finite p-groups in (Huppert 1967). Around 1970 various authors, perhaps prompted by read ing this account of the theory, took up the study these groups and substantial advances were made in understanding their structure (Shepherd 1971; Miech 1970, 1974a,b; Leedham-Green and McKay 1976b, 1978a,b, 1984).
Pro-p-groups
Authors: Leedharn-Green C R , McKay S
Published: November 2023

This chapter begins the investigation of the connection between pro-p-groups of coclass r and finite p-groups of coclass r. The first three sections of the chapter contain the basic results and definitions in the theory of pro-p-groups that will be needed, beginning with more general topological groups. Again standard results are often quoted without proof, and can be found in (Pontrjagin 1957) or in (Dixon et al. 1999). See also (Wilson 1998).
Connections
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

In this chapter we begin our study of Yang-Mills theory—the theory of connections-which makes up the core of this book. Classical differential geometry considers, by and large, connections on the tangent bundle of a manifold, the most important being the Levi—Civita connection defined by a Riemannian metric. The key feature of Yang—Mills theory is that it deals with connections on auxiliary bundles, not directly tied to the geometry of the base manifold. After reviewing standard definitions and notation in Section 2.1, we go on to give proofs of two fundamental theorems: the integrability theorem for bolomorphic structures and Uhlenbeck’s theorem on the existence of Coulomb gauges. These are essentially local results. We regard them as extensions of the elementary fact that connections with curvature zero are locally trivial.
Yang-Mills Moduli Spaces
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

In this chapter we introduce moduli spaces of ASD Yang-Mills connections and develop some of their basic properties. Let E be a bundle over a compact, oriented, Riemannian four-manifold X. At the level of sets, the moduli space ME is defined to be the set of gauge equivalence classes of ASD connections on E: that is, we identify connections which are in the same orbit under the action of the bundle automorphism group 𝒢 The moduli space has a natural topology, induced by the topology in the space 𝒜 of connections.
Stable Holomorphic Bundles Over KÄhler Surf Aces
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

In this chapter we consider the description of ASD moduli spaces over complex KÄhler surfaces. This discussion takes as its starting point the relation between the ASD equation and the integrability condition for § operators which we have seen in Chapter 2. We show that the ASD connections can be identified, using a general existence theorem, with the holomorphic bundles satisfying the algebro-geometric condition of ‘stability’. Examples of concrete applications of the theory have been given in Chapter 4; and in Chapters 9 and 10 we will see how these ideas can be used to draw conclusions about the differential topology of complex surfaces.
Non-Existence Results
Authors: Donaldson S k. , Kronheimer P B
Published: November 2023

In this chapter we return at last to address some of the problems in fourmanifold topology which were raised in Chapter 1. We will prove Theorems (1.3.1) and (1.3.2) on the non-existence of manifolds with certain intersection forms. Throughout the chapter, the strategy of proof of these non-existence results will be the same. Supposing that a manifold X of the type in question exists. one chooses judiciously a bundle E over X and studies the moduli space of ASD connections M=ME. From the known topological features of the moduli space and its embedding in the space ℬE one derives a contradiction.
Generalized Riemann And Variational Integration In Division Systems And Division Spaces
Authors: Henstock Ralph
Published: November 2023

My thanks are due to Thomson (1971a, b, c, 1972a, b, 1975), who examined the theory and showed that it would work with fewer hypotheses, coining the name, division system, which in this book is called a free division system. Here I have called a division system one that considers divisions of a fixed elementary set. See Example 1.1.3 for some details of Thomson’s system.
Integration In Infinite Dimensional Spaces
Authors: Henstock Ralph
Published: November 2023

In countable Cartesian product spaces (sequence spaces) a result of Jessen (1934) holds, so that the integral is the limit of the corresponding integral over the first n terms of the sequences. The origins of integration in uncountable Cartesian product spaces (function spaces) are in Einstein (1906) and Smoluchowski (1906), examples being given by Chandrasekhar (1943). Two integrals vital for theoretical physicists and theoretical chemists are in Wiener (1930) and Feynman (1948). It may be that the integral of this chapter gives mathematical expression to the latter; other expressions used are far too complicated.
Functional Analysis And Integration Theory
Authors: Henstock Ralph
Published: November 2023

A sequence satisfying (8.2.31) is called fundamental or Cauchy, one satisfying (8.2.33) is called convergent to f, and the theorem shows that every fundamental sequence is convergent, and the function space is complete.
Introduction
Authors: Dwork Bernard
Published: November 2023

We examine the Boyarsky principle for generalized hypergeometric functions. This involves understanding the integral representations of such functions to the point of being able to show that the Frobenius matrix varies analytically with the multiplicative parameters (e.g. a, b, c in the case of the Gauss 2F1(a, b, c, λ). Following a suggestion of G. Laumon (letter of 25 April 1983) we use the Laplace transform to develop a theory broad enough to encompass the four hypergeometric functions of Appell (Appell and Kampe de Feriet 1925), their Lauricella generalizations and kFk-i (Erdelyi et al. 1953). Even in the case of 2F1 the present work represents a significant improvement over our previous study (Dwork 1983) of the Boyarksy principle. From the point of view of cohomology associated with hypergeometric functions our work represents an improvement over that of Aomoto (1975, 1977) who seems to have restricted his attention to cyclic coverings of ramified at a set of non-singular hypersurfaces with normal crossings (i.e. the intersection of each subset is a non-singular complete intersection). The condition is satisfied by 2F1, F; of Appel and the Lauricella FD but not by the remaining hypergeometric functions listed above.
Multiplication By xu (Gauss Contiguity)
Authors: Dwork Bernard
Published: November 2023

Let n1, n2, d1, …, d2 be strictly positive integers. Let H be the hyperplane in n = n1 + n2 affine space of characteristic zero defined by
Deformation Theory
Authors: Dwork Bernard
Published: November 2023

We use the notations of Chapters 2 and 3 except that here λ= (λ1, …, λm) consists of m algebraically independent elements over ℚ(π, a). We put B = ℤ[λ], K = ℚ(a, π, λ) and completion will play no role. We consider satisfying (2.6).
Linear Differential Equations Over a Ring
Authors: Dwork Bernard
Published: November 2023

Let A be an integral domain with quotient field K of characteristic zero. Let ϕ, ∈ A[s] be a polynomial in one variable with coefficients in A. For each s ∈ ℤ
Singularities (Generalities)
Authors: Dwork Bernard
Published: November 2023

We now discuss the singularities of eqn (5.17) . To clarify the notation we emphasize the tangent space aspect of the deformation mapping. In this direction let a denote a generic point on H over ℚ, let λ = (λ1, …, λm) be a generic point in affine m-space and Γ a generic point on the affine line.
Non-Regular Case
Authors: Dwork Bernard
Published: November 2023

We shall use the notation of Chapters 2 and 3. We here put B0 = ℤ[λt, …, λm] and B = B0[Γ] so that our emphasis shall be upon variation with Γ. In particular the λi may be viewed as specialized. We suppose that K is a field containing B0 and all necessary elements of H, but that Γ is transcendental over K. We shall consider R^ the K-space spanned by {Xu}u∈H^0 and R^Γ=K(Γ)⊗R.
Modified Laplace Transform
Authors: Dwork Bernard
Published: November 2023

This chapter contains the motivation for this book, but the notation will be independent of the rest of the book. Let K be a field of characteristic zero.
Algebraic Theory of Laplace Transform
Authors: Dwork Bernard
Published: November 2023

The object of this section is to show that subject to modest restrictions, the Laplace transform gives an isomorphism of Wa′ with a suitable factor space of ℛ0′, the space of elements of K[X1,…,Xn11/X1…Xn1f(1)…f(n2)] which as rational functions of X1,…,Xn1 are homogeneous of degree zero. The notation of this section will coincide with that of Chapter 2. For this purpose we rewrite (10.8) so as to confom1 with that notation. We recall (2.7)
Examples
Authors: Dwork Bernard
Published: November 2023

The object of this section is to construct the ring R^ corresponding to various classical hypergeometric functions and to verify hypotheses (2.4), (2.5) and (2.6). To simplify notation we shall drop all π factors. By Proposition 6.13.1 the explicit verification of (2.5) may be omitted.
Duality-Analytic Theory
Authors: Dwork Bernard
Published: November 2023

The object of this section is to explain the relation between the mappings 0 (17.9) and αa,a′,λ∗of (4.21).
An Historical Introduction
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

We shall likely never know who first noticed that given seven elements, seven sets of three elements each can be found so that every pair of two elements appears in exactly one of the seven chosen sets; or who first noticed that writing down the rows, columns, forward diagonals, and back diagonals of a 3 × 3 matrix of nine distinct symbols yields twelve sets of three elements chosen from nine, again so that every pair of two elements appears in exactly one of the twelve sets.
Existence: Direct Methods
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Constructions for designs fall into two main classes. Directconstructions produce a design without requiring a (smaller) design of the same type in the construction, while recursiveconstructions produce larger designs from smaller ones of the same type. In this chapter, we examine direct constructions for triple systems. Then in Chapter 3, we turn to recursive methods.
Enumeration
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

In §0.4 and §0.5, some history of hand computations of catalogues of triple systems is outlined. In this chapter, we give catalogues of triple systems for small orders. Each block is presented vertically. The order of the smallest nontrivial Steiner triple system is 7; a system is given in Table 5.2. This design is unique up to isomorphism, and is the design of points and lines of PG(2,2), the projective plane of order 2.
Automorphisms I: Small Groups
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

LetVbe a set of velements, and let Γ G be a permutation group acting on V. Γ G is a set of permutations containing the identity permutation and closed under composition of permutations. Thus in order to specify the group Γ, it suffices to give a set (π1, ... , πr) ⊆ Γ of generators,so that every π ∈ Γ can be written as a finite sequence of compositions of the generators.
Automorphisms II: Large Groups
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

In this chapter, we study triple systems admitting ‘large’ automorphism groups. In particular, we examine actions of permutation groups on triple systems that are, in some way, uniform.
Leaves and Partial Triple Systems
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

A partial triple systemof order vand index λ or PTS(v,,\), is a collection of triples on velements, so that no pair occurs in more than λ of the triples. Often the parameter λ is omitted when λ = 1. A PTS is also called a packingin the literature. A rnultigraph on vvertices is a A-leaveif and only if there exists a partial triple system of order vand index λ whose leave is the specified multigraph. For example, the 4-vertex multigraph K1,3 with edge set ( (1, 2), (1, 3), (1, 4)) is a I-leave, but is not a 2-leave. In this chapter, we examine the question of which multigraphs are λ-leaves.
Embedding and its Variants
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

In this chapter, we consider embedding (partial) triple systems into triple systems. Given a partial triple system (V, B) of order ν and index λ, when is there a triple system (W, D) of order wand index ³such that V<Wand Bis a submultisct of D? Such a containing system (W,D) is an enclosingof (V,B).When, λ= μ the enclosing is an embedding. when v= w,the enclosing is an immersion.When both equalities hold, the enclosing is a completion.
Neighbourhoods
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

We view the neighbourhood of an element xas a multigraph on the remaining v−1 elements, whose edges are the pairs appearing in triples with x.In general, a multigraph is a λ.-neighbourhoodif it is the neighbourhood of some element in a triple system of index, λ The basic question is to determine which multigraphs are neighbourhoods.
Configurations
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

By a configuration we mean a PTS(ν, λ.), typically having a fixed small number of triples. The triples are sometimes called lineshere to conform with geometric terminology (in the same vein, elements are sometimes called pointshere). The degreeof a point is the number of lines containing the point.
Large Sets and Partitions
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Kirkman (1850) exhibited the first large set of triple systems, an LTS(9,1). Sylvester (1861,1893) later studied the same case, as did Bays (1917) and Emch (1929).
Chromatic Index and Resolvability
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Determining the size of a maximum PPC in an arbitrary TS(v, λ.)appears to be a more difficult question. The first nontrivial bound on the size of a maximum PPC in STSs was established by Lindner and Phelps (1978); it says that every STS(v) contains a PPC of size at least (v−1)/4 for vsufficiently large.
Orthogonal Resolutions
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

The motivation for studying orthogonal resolutions of TS(v,,λ)s (or, more generally, of designs) comes from their connection to the existence of various kinds of square and (multidimensional) cubical arrays.
Stss With Two Subsystems
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

Colbourn, Oravas, and Rees (1999) address the existence of Steiner triple systems of order vthat contain two subsystems, one of order uand the other of order w,which are either disjoint, or share precisely one element. This problem can be cast in quite general terms as follows.
Mendelsohn Triple Systems
Authors: Colbourn Charles J , Rosa Alexander
Published: November 2023

In this chapter, we consider a second type of ordering: designs in which the blocks are cyclic triples. A Mendelsohn triple systemof order vand index λ is a pair (V, B). Vis a v-set of elementsand Bis a collection of cyclic triples on V;every ordered pair of distinct elements from Vappears in precisely λ of the cyclic triples of B.The abbreviation MTS(ν, λ) is typically employed, and the parameter Ais often omitted when λ= l. Mendelsohn (1971) first studied such designs under the name ‘cyclic triple systems’, which unfortunately can be confused with triple systems having a cyclic automorphism.
Fundamentals
Authors: Higgins Peter M
Published: November 2023

The purpose of this chapter is to introduce the basic concepts and theorems of algebraic semigroups, together with the additional ideas that are to be developed in this monograph. Its content forms a concise course in algebraic semigroup theory suitable for the graduate or research student, or as a supplement to an advanced undergraduate course in this subject. Proofs will be provided for all facts relevant to our purposes, although some simple results will be set as exercises or left to the reader to supply the details. There are several other well-known texts containing broadly similar material, the most suitable for undergraduates being Howie (1976), especially the first five chapters. The texts of Clifford and Preston (1961, 1967) provide a particularly careful and thorough account of the work done up until their time of publication, while an early Russian text available in English is that of Lyapin (1974). Other well-known books on the subject include those of Petrich (1973, 1977, 1984) and Lallement (1979), which concentrates on applications to coding and language theory. Also along these lines, Pin (1986) is noteworthy in giving an introducton to the theory of automata and formal languages, together with many worked examples and interesting glimpses of current research topics. Howie (1991) treats similar material, but with the undergraduate audience in mind.
Zigzags and their applications
Authors: Higgins Peter M
Published: November 2023

We begin this chapter with a new and short proof of the Zigzag Theorem of John Isbell, which characterizes semigroup dominions by means of certain sequences of equations known as zigzags. A variety of applications concerning epimorphisms and amalgams will illustrate the usefulness of this result. First we look at the connection between epimorphisms and dominions. Observe that to say that a homomorphism α:S →T is epi (pre-cancellative) is the same as saying that whenever homomorphisms β, γ: T → V are such that β | Sα= γ |Sα, then β = γ. If this is the case, we say that Sα isdense in T, or that Sα isepimorphically embedded in T via the inclusioni: Sα →T. Therefore the study of epimorphisms is equivalent to the study of epimorphic embeddings.
Combinatorial aspects of transformation semigroups
Authors: Higgins Peter M
Published: November 2023

In this section we shall introduce results of a probabilistic nature concerning finite transformation semigroups. The study of random functions has a wide literature of its own, and many questions along these lines lead to occupancy problems which arise throughout mathematics and physics, particularly in mathematical genetics (Kingman 1978). We concentrate here on results that are of interest to semigroupists and which have elementary proofs. Indeed, one justification for the inclusion of this material is the delightful manner in which classic results from analysis and combinatorics furnish answers to a number of natural questions concerningTn andOn (the semigroup of all order-preserving mappings on {l, … , n}).
Modules, they are also representations
Authors: Helemskii A YA
Published: November 2023

Catherine the Great, “Tell me, General, what exactly is the difference between a howitzer and a mortar?” Cossack-General Platov, “Oh, you see, Mother-Czarina, a howitzer thing and is a specialized sort of this is a mortar - an altogether specialized sort of thing!” (From an old Russian anecdote.) The reader has already been introduced to representations and he can judge from the results of Chap.IV §§5-7 how useful they can be. When we consider the representations of a given algebra it is as though we were taking photographs of it from different sides. The “operator” photos we get (which may also include, as in Theorem IV.7.57, three-dimensional photos) enable us to get a much better understanding of its structure and to form a foundation for its study using results and methods accumulated in operator theory.
Problems leading to homology and the elements of homology theory
Authors: Helemskii A YA
Published: November 2023

We have already met the word “homology” and moreover (like “algebra” and “topology”) in two senses: as a specific mathematical object (0.4.22) and as a branch of mathematics. This last chapter is devoted to those concepts and results in the theory of “the algebras of analysis” that are just as much part of the subject of homology — or that in each case can be interpreted in typical homological terms. Many of the concepts introduced here, for example projective modules, derivations and extensions, seem at first to be completely unconnected. Gradually, however, some connection between the separate subjects begins to emerge (cf. 2.40) and finally, just before the final curtain, it is shown (§3.3) that they may be considered in the context of a certain unified approach and studied by general methods
The norm map
Authors: Evens Leonard
Published: November 2023

Let G be a group and H a subgroup of finite index. Let Φ : G → S ∫ H be the monomial representation defined in Section 5.2. Here we can take S = 𝒮(G / H) or any subgroup of 𝒮(G / H) which contains the image of π : G → 𝒮(G / H) (the representation of G as a group of permutations of its left coset space G / H). In particular, if H is normal in G, then that image is isomorphic to the factor group G / H itself.
Spectral sequences
Authors: Evens Leonard
Published: November 2023

Spectral sequences play an important role in group cohomology because they provide a means of reducing cohomology in a complex situation to the cohomology of constituents. The most important spectral sequence for us will be the Lyndon–Hochschild–Serre (LHS) spectral sequence which relates the cohomology of a group to that of a normal subgroup and that of the factor group. We have seen two special cases of this situation. The Kiinneth Theorem relates the cohomology of K × H to that of K and H; in particular if k is a field
Some related theorems
Authors: Evens Leonard
Published: November 2023

We round off our discussion of varieties and complexity with some consequences of the results in the previous two chapters and some other results employing ideas from commutative algebra.
Local singular series
Authors: Effinger Gove W , Hayes David R
Published: November 2023

Let Fqbe the finite field of q elements, let T be transcendental over Fq, and let k be the rational function field Fq(T). We assume that the reader is familiar with the concepts of discrete valuations of a field and the associated ‘places’ of the field (cf. Borevich and Shafarevich 1966, Serre 1979).
Local Gauss sums and local derivatives
Authors: Effinger Gove W , Hayes David R
Published: November 2023

In the first two sections of this chapter, we define local Gauss sums as integrals in Haar measure, and we prove some basic theorems evaluating their absolute values. The methods of proof are classical. In the last two sections, we apply these evaluations to compute some local Radon Nikodym derivatives. The notations introduced in Chapter 2 remain in force.
The adèle ring over k
Authors: Effinger Gove W , Hayes David R
Published: November 2023

In the classical Hardy-Littlewood ‘circle method’, partition problems in Z are solved by integrating appropriate generating functions over the unit circle in the complex plane. For the study of partition problems in A, the most convenient analogue for the circle is the adele class group. The complex path integral used by Hardy-Littlewood is replaced by the Haar integral on AK/K.
L-functions of Dirichlet type
Authors: Effinger Gove W , Hayes David R
Published: November 2023

In classical number theory, the analysis of problems involving prime mun hers often leads to questions about the Riemann zeta-function or its genera lizations, the Dirichlet £-functions. In this chapter, we develop a particular class of £-functions over k and use them to prove a bound for character sums (Theorem 5.7) which will be of fundamental importance in our anal ysis of the polynomial 3-primes problem.
Tauberian theorems − I
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

We begin by restating some results for Borel’s methods that have been stated in chapter 3. As we develop the results in this chapter, it will become clear how these results are the origins of the more general theorems. These theorems are of three types.
Relationships with other methods
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

9.1 Product methods with the Cesàro method

We recall that (10.1)Ank = ∑r=0nErkan−r
Classical Objects
Authors: Yves Diers
Published: November 2023

In an arbitrary Zariski category A, we define a range of objects corresponding to the usual types of commutative ring: To any type of objects there corresponds a type of morphism in that enables categories to be obtained and the usual constructions to be performed as universal constructions. Different types of congruence occur naturally with the different types of object and correspond to usual ideals: proper, maximal, prime, radical, quasi-primary, primary, and irreducible.
Jacobson Ultraschemes
Authors: Yves Diers
Published: November 2023

An object A in a Zariski category is a Jacobson object if any prime congruence on A is the meet of maximal congruences. The prime spectrum of a Jacobson object is a Jacobson space, i.e. a space in which any closed set is the closure of the set of its closed points. Thus the maximal spectrum of a Jacobson object is very dense in its prime spectrum, and for many purposes it can replace the prime spectrum. Jacobson schemes are obtained by gluing together affine schemes of Jacobson objects, while Jacobson ultraschemes are obtained by gluing together their affine ultraschemes. The ultraschemes are introduced in order to remove the generic points and deal only with the closed points.
Neat Objects And Morphisms
Authors: Yves Diers
Published: November 2023

Neat objects are introduced in order to describe for arbitrary objects in a Zariski category A the analogue of the separability property for field extensions. Therefore, for a finite field extension k → K, the object K in CAlg ( k) will be neat if and only if K is a separable extension of k. The classical definition of neat algebras by means of nilpotent ideals cannot be used in an arbitrary Zariski category because of lack of such ideals, for example, in the category RedCRng of reduced rings. We shall use another feature of the separability property, namely that two roots of a separable irreducible polynomial P∈k [X] in a local k-algebra Lare either equal or relatively prime. If K is the splitting field of P, this property is equivalent to the property that any pair of morphisms (f, g) : K ⇉ L in CAlg ( k) is either equal or codisjointed. Thus we have the definition of a neat object: a finitely presentable object A such that any pair of morphisms (f, g) : A ⇉L into a local object L is either equal or codisjointed. Indeed, there are many other equivalent definitions of neat objects, including the following: a neat object is a finitely presentable codecidable object, i.e. an object whose codiagonal is a direct factor. Naturally, in any category of commutative algebras CAlg (A) over some ring A, neat objects are precisely classical neat algebras.
Etale Objects And Morphisms
Authors: Yves Diers
Published: November 2023

An etale morphism in a Zariski category A is a morphism f: A→ B whose spectrum Spec(f): Spec(B)→ Spec(A) makes Spec(B) a kind of etale space over Spec (A). Because Spec (A) and Spec (B) are not mere topological spaces but modelled spaces, the usual definition of an etale continuous map cannot be used as such. Nevertheless, a close look at the universal properties of etale continuous maps in categories of topological spaces will help us to understand etale morphisms in A. In the meantime, we have to bring in to question the fact that Spec(f) must be a local homeomorphism. It is easy to go from etale objects to etale morphisms, and vice versa, by identifying a morphism/: A →B in A with an object in the Zariski category A/ A.
Terminators
Authors: Yves Diers
Published: November 2023

The idea of terminators is similar to that of codisjunctors, except that pushouts are used instead of coequalizers. In an arbitrary category A with a strict terminal object denoted by 1, let f: A → B be a given morphism. A morphism g : A → C is said to terminate f if the pushout of (j, g) is terminal, i.e. the pushout object is 1. In categories of commutative algebras, to any object C is associated a universal van Neumann regular object nc : C → N ( C) and it turns out that a morphism g : A → C terminates f if and only if the morphism neg: A → N ( C) does. Therefore the study of morphisms terminating f can be restricted, to some extent, to those morphisms g : A → C which have a van Neumann regular codomain and are called van Neumann morphisms. A terminator off is then defined as being a universal van Neumann morphism terminatingf, i.e. a van Neumann morphism terminatingf and such that any van Neumann morphism terminating f factors through it in a unique way.
Necessary conditions for topological stability
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

The object of this chapter is to show how the general theory of Chapter 5 is applied in order to obtain necessary conditions for 𝒞 0-stability in terms of multitransversality to certain submanifolds of jet space. These (‘LST-invariant’) submanifolds are locally unions of ST-equivalence classes satisfying a fairly mild ‘immersion condition’. Our first main result (Theorem 6.2.9) shows that in dimensions where such a submanifold generically has a non-empty presentation, a map-germ not transverse to it is disruptive. The results of Chapter 5 then imply that transversality to such a manifold is a necessary condition for local 𝒞 0-stability, 𝒞 0-stability (if the source is compact), and so on.
Stable topological type of finite mapgerms
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

This chapter forms a companion to chapter 7. Its objective is to classify a substantial class of germs up to some convenient equivalence relation (contact equivalence, alias 𝒦-equivalence, where this is feasible) and then to re-examine the classification from the viewpoint of ST-equivalence. We recall from chapter 6 that two germs are stably topologically equivalent (ST-equivalent for short) if they have stable unfoldings which are equivalent by homeomorphisms of source and target. A stratum defined by a 𝒦-invariant submanifold of some jet space is LST-invariant if the submanifold has a neighbourhood N such that germs with jet in the stratum are ST-distinct from others with jet in N. We define fine LSTinvariants as in Section 6.5; the definition is recalled in the introduction to Chapter 7. The background to these definitions, their importance, and the way they are applied to obtain necessary conditions for topological stability, are explained in Chapter 6. Applications of the results of this chapter to obtain sufficient conditions for topological stability are given in Chapters 10 and 11.
Algebraic criteria for civilization
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

In this chapter we describe methods for establishing the transversality conditions required to prove, via Theorem 9.6.6, that submanifolds of jet space are civilized.Our approach is based on the following division of the target of a smooth map f: N → P into (c  ∞); -stable and unstable points. We say that y  ∈  P is a (c∞ -) stable point if the germ off at ∑.(f, y) = ∑.(f) ∩f-1(y) is a c∞ =-stable germ (hence in particular if ∑..(f, y) = Ø); otherwise y is an unstable point. We will call the set of unstable points the instability locus. Observe that, if f |∑.(f) is proper, this is always a closed set.
Elliptic partial differential equations in the worst case setting
Authors: Werschulz Arthur G
Published: November 2023

In this chapter, we use the techniques of information-based complexity to find algorithms that are optimal in the worst case setting for elliptic partial differential equations (PDE).
The average case setting
Authors: Werschulz Arthur G
Published: November 2023

In this chapter, we change our point of view, considering our problems in an average case setting, rather than a worst case setting. Why do we wish to go to a new setting? One could always appeal to sheer mathematical interest. However, there is a simple, pragmatic justification for leaving the worst case setting, namely, that many of the problems we want to solve are too hard under the worst case setting. We found that elliptic PDE, elliptic systems, and the Fredholm problem of the second kind were intractable, in the sense that their complexity was exponential in the dimension of the domain. Moreover, those problems are nonconvergent if the problem elements are only smooth enough that existence and uniqueness of solutions hold. Things were even worse for ill-posed problems, which we found to be strongly noncomputable in the sense that the cost of finite error approximation is infinite, no matter how large we allow the error to be.
Preliminaries on Mechanics of Continuous Media
Authors: Ionescu Ioan R , Sofonea Mircea
Published: November 2023

The object of this chapter is to summarize some basic concepts and general principles of continuous mechanics. To this end, elements of kinematics, the balance laws, the stress tensors, and the constitutive laws in the study of continuous media are presented. All the functions involved here are assumed sufficiently smooth and the classical notations ▽, div, Div for the gradient operator, the vectorial divergence operator, and the tensorial divergence operator are used.
Quasistatic Processes for Rate-Type Viscoplastic Materials
Authors: Ionescu Ioan R , Sofonea Mircea
Published: November 2023

In this chapter we study quasistatic problems for semilinear rate-type constitutive equations. The existence, uniqueness, and finite-time stability of the solution are given using some arguments of ordinary differential equations in Hilbert spaces. The case of viscoelastic materials is also investigated with respect to asymptotic stability, existence of periodic solutions, approach to non-linear elasticity, and long-term behaviour of the solution. By passing to the limit, the elastic-perfectly plastic case is derived. A Euler method, internal and external approximation techniques are used in order to reduce the continuous problem to a sequence of linear algebraic systems. Existence and uniqueness results in quasistatic processes for rate-type constitutive equations involving a hardening parameter are also given.
Elliptic Operators and Lie Groups
Authors: Robinson Derek W
Published: November 2023

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

Both the homogeneous and the inhomogeneous heat equations have a number of geometric and analytic properties which extend their interest beyond the problems of thermodynamics. The geometric properties arise because the equations reflect the physical observation that heat flows from warmer regions to colder regions. The analytic properties are a consequence of the smoothness of the heat redistribution process.
More geometry for Hering’s 3: SL(2, 13)
Authors: Buekenhout F
Published: November 2023

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

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

“Most” of the known families of finite generalized quadrangles arise by this procedure (see Kantor (1980); Payne (1990)). It seems likely that Q must be a p-group for some prime p. While this remains open, there has been significant progress (Frohardt (1988); Chen and Frohardt (submitted)).
Collineation groups of (q + t1 t)-arcs of type (01 21 t) in a Desarguesian plane of order q
Authors: Korchmáros G
Published: November 2023

The known examples have a common property of having a tnucleus in the sense that their t-secants are concurrent at a point.
A geometric approach to semi-cyclic codes
Authors: Maruta T
Published: November 2023

Throughout this paper, a code means a linear code over the Galois field F = GF(q) of order q with the minimum distanced ≤: 3.
Remarks on non-associative Galois theory
Authors: Mason G
Published: November 2023

In this paper we discuss the possibility of establishing similar results in more general contexts, mainly concerning translation planes. Here K becomes the (non-associative) co-ordinatizing ring of the plane, k is the kernel, and G a group of automorphisms in the so-called linear translation complement — that is to say G fixes k pointwise.
Codes of nets with translations
Authors: Moorhouse G E
Published: November 2023

Theorem 2. Conjecture 1 holds for 4-nets constructed from 3xp difference matrices over cyclic groups of prime order p, that is, for 4- nets of prime order p admitting a central translation of order p.
Internal nuclei of k-sets in finite projective spaces of three dimensions
Authors: Wettl F
Published: November 2023

It is well known that an oval, that is a (q + 1)-arc in an even order plane has a nucleus, that is a point such that every line through this point intersects the oval in exactly one point. Every point of a (q + 2) arc (hyperoval) is a nucleus of the remaining points.
Menon difference sets and relative difference sets
Authors: Wild P R
Published: November 2023

The only known cyclic Menon difference set is the trivial difference set in Z4 corresponding to the Barker sequence 111-1. It has been conjectured that no other cyclic Menon difference set exists.
Prime decomposition
Authors: Klingen Norbert
Published: November 2023

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

Arithmetical equivalence which we are going to study in this chapter imposes a much stronger relationship between prime decomposition in different number fields than Kronecker equivalence did.
Variations
Authors: Klingen Norbert
Published: November 2023

In this chapter we report on generalizations and refinements of the theory developed in the preceding chapters, as well as on results from related areas which use the same methods or lead to similar group theoretic problems. So this chapter shows the range of questions from number theory, algebra and even from other parts of mathematics which lead to group theoretic questions as discussed in this book. This chapter can be understood as being a guide to the literature and providing some hints for possible further investigations in this area.
Applications To Homology 3-Spheres
Authors: Petrie Ted , Randall John
Published: November 2023

We now establish, after some preliminary details, a one-to-one correspondence between restricted G-line bundles over M and restricted line bundles over DX. First we show that restricted line bundles over DX determine unique restricted G-line bundles over M.
Using SHEEP and CLASSI
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

This section explains how to use LOAD to read data and source code, and how to adapt LOAD’s search path to look through additional directories. The LOAD command can take (at least) two different forms.
The exterior calculus package EXCALC
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

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

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

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

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

Algebraic and sheaf-theoretic methods have been used to solve a series of classical problems in the geometry of complex manifolds and in the theory of functions of several complex variables. They have also oriented the research related to several complex variables for a good few decades. Developed in parallel with Grothendieck's theory of schemes, the theory of complex analytic spaces represents nowadays the natural framework and the language in which these algebraic methods are formulated.