bluuebunny/crossref_metadata_2025_split huggingface dataset:
Non-Regular Case
Authors: Dwork Bernard 
Published: November 2023

Abstract
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.
Algebraic Theory of Laplace Transform
Authors: Dwork Bernard 
Published: November 2023

Abstract
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)
An Historical Introduction
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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.
Leaves and Partial Triple Systems
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
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

Abstract
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&lt;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

Abstract
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.
Chromatic Index and Resolvability
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
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.
Stss With Two Subsystems
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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
Local singular series
Authors: Effinger Gove W , Hayes David R 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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.
Classical Objects
Authors: Yves Diers 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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.
The average case setting
Authors: Werschulz Arthur G 
Published: November 2023

Abstract
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

Abstract
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

Abstract
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.