bluuebunny/crossref_metadata_2025_split huggingface dataset:
Scattered Data Surface Fitting
Authors: Dierckx Paul 
Published: November 2023

Abstract
In this chapter we will present algorithms for fitting a bivariate tensor product spline to data values given at arbitrary points in a rectangular approximation domain. First we will consider the least-squares criterion and explain how we can exploit the properties of B splines to obtain an observation matrix with minimal bandwidth. We will also discuss a number of methods for obtaining a unique solution in the case of rank deficiency and illustrate how we can create surfaces with special features by using knot curves instead of constant knots. Then we will explain how to determine bivariate smoothing splines. For this we will derive a suitable smoothing norm for tensor product splines and generalize the one-dimensional knot placing strategy.
Cauchy’s Formula and Its Applications
Authors: Needham Tristan 
Published: November 2023

Abstract
One of the principal objectives of this brief chapter is to tie up various loose ends from previous chapters. In particular, we have previously claimed (but have not yet explained) three important properties of an analytic function f(z):We can differentiate /(z) as many times as we please—it is “infinitely differentiable”.In the vicinity of an ordinary point, f(z) can be expressed as a Taylor series.
Vector Fields: Physics and Topology
Authors: Needham Tristan 
Published: November 2023

Abstract
Throughout the course of this book we have relied on a singlemeans of visualizing a complex function, namely, as a mapping of points in one complex plane to points in another. This idea has proved to be extremely powerful, for in terms of it the complex derivative is nothing more complicated than a local amplitwist. Despite its many virtues, in this chapter we shall abandon the mapping paradigm and introduce a completely new one in its place, thereby gaining a host of fresh insights into the subject and revealing surprising connections with physics.
Area and volume
Authors: Roe John 
Published: November 2023

Abstract
The concept of area is sometimes introduced by talking about the ‘amount of paint’ required to cover a certain figure in the plane. Thus a square of side 2 feet would need four times as much paint to cover it as a square of side 1 foot, a circular disc of radius 1 foot would need a little over three times as much, and so on. This simple idea of the ‘amount of paint’ suggests to us several properties that area should have, which we could use as axioms; for example, the property that the area of a figure made up of two separate pieces is the sum of the areas of the individual pieces. However, if our theory is to be precise, we need to start one stage further back, and to say what kinds of ‘figures’ we intend to find the area of.
An application to number theory: prime integer points
Authors: Huxley M N 
Published: November 2023

Abstract
A favourite source of problems in number theory is to take two integer sequences, and to ask how many members they have in common. Popular sequences are the prime numbers, the Fibonacci numbers, the powers of 2 and the squares (or, more generally, the values of a suitable polynomial g(x)). Problems of this type involving prime numbers are tantalizingly difficult: even primes of the form n 2 + 1 are well out of reach.
A survey of Jordan groups
Authors: Dugald Macpherson 
Published: November 2023

Abstract
This article surveys recent results on a rather ancient topic in permutation group theory. In the 1870s, Jordan developed the early theory of the permutation groups now known as Jordan groups, that is, permutation groups in which, in some non-degenerate way, the pointwise stabilizer of a subset is transitive on the complement. One such reference is Jordan (1871). Over the last 20 years a much clearer picture of them has developed. Finite primitive Jordan groups were classified in the early 1980s. In a slightly vague sense, there is now a classification of infinite primitive Jordan groups (Adeleke and Neumann, in pressb; Adeleke and Macpherson, 1994). Some of these results, in particular the classification of finite primitive Jordan groups, have had important applications in model theory, and model-theoretic methods have been used to construct new Jordan groups. The purpose of this paper is to describe the known examples of Jordan groups and the classification theorem, and to sketch some of its applications to permutation group theory and to model theory. I also consider how some of the relational structures associated with Jordan groups arise in other contexts. The article is essentially a survey of known results.
Symmetric Cones
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
This chapter is elementary; we use only the geometry of finite dimensional Euclidean spaces. A large part of the chapter is about proper convex cones, which form a much more general class than symmetric cones. For a number of questions, such as the Laplace transform and Fourier-Laplace transform considered in Chapter IX, this is the natural generality.
Introduction
Authors: Rahman Matiur 
Published: November 2023

Abstract
The main topics of this book relate to surface waves in oceans and tidal waves in estuaries. A proper understanding of these phenomena is crucial to much of physical oceanography and is an exciting area of study. Many scientific papers have appeared in learned journals in which mathematical models have been used to correlate predicted and experimental data. Perfect correlation is the ultimate goal of the mathematical models, and although much has been achieved, there is great scope for future work.
Nonlinear long waves in shallow water
Authors: Rahman Matiur 
Published: November 2023

Abstract
In Chapter 5 the Stokes perturbation procedure for finite-amplitude waves was developed. In that method the perturbation parameter was used as kH /2, the relative wave steepness. However, in very shallow water (see Chapter 5) the Stokes perturbation techniques fails to hold good which will be clear from the following analysis.
Introduction To Wavelet Analysis Over
Authors: Holschneider M 
Published: November 2023

Abstract
Wavelets comprise the family of translations and dilations of a single function called the wavelet. The name is due to the fact that every wavelet has to have at least some oscillations, and that it is localized. The wavelet transform is the set of scalar products of all dilated and translated wavelets with an arbitrary analysed function. This linear map is a partial isometry and its range is characterized by a reproducing kernel.
Symmetric Functions and Hall Polynomials
Authors: Macdonald I G 
Published: November 2023

Abstract
This is a new and much expanded edition of Professor Macdonald's acclaimed monograph on Symmetric Functions and Hall Polynomials. Almost every chapter has new sections and many new examples have been included throughout. In addition there are two new chapters (6 and 7). Chapter 6 contains an extended account of a family of symmetric functions depending on two parameters. These symmetric functions include as particular cases many of those encountered earlier in the book and they also include, as a limiting case, Jack's symmetric functions depending on a parameter a. Many of the properties of the Schur functions generalize to these two-parameter symmetric functions. Chapter 7 is devoted to the study of the zxonal polynomials, long familiar to staticians. From one point of view, they are a special case of Jack's symmetric functions (the parameter a being equal to 2) but their combinatorial and group-theoretic connections make them worthy of study in their own right.
Symmetric Functions With Two Parameters
Authors: Macdonald I G 
Published: November 2023

Abstract
Another example is provided by the zonal polynomials (Chapter VII). Up to a scalar factor they are characterized by the same properties (a) and (b), but with a different choice of scalar product, namely More generally, Jack’s symmetric functions (§10 below) are characterized by (a) and (b), the scalar product now being defined by where a is a positive real number. They reduce to the zonal polynomials when a - 2, and to the Schur functions when α- 1.
Grassmann Geometries
Authors: Pasini Antonio 
Published: November 2023

Abstract
In this chapter we show how to construct geometries with string diagram graphs starting from geometries with arbitrary diagram graphs. This will allow us to extend to geometries with non-string diagram graphs the concepts defined for geometric graded posets in Sections 4.3.2 and 4.3.3. How ever, in order to explain that construction we need to state some lemmas on topological properties of sets of vertices of graphs. We will do this in the next section.
Universal Covers
Authors: Pasini Antonio 
Published: November 2023

Abstract
We should now prove the existence of universal covers. That is, we should prove that, for every geometry Γ, there is a geometry (uniquely determined up to isomorphism) which is the universal m-cover of Γ. Actually, we can prove this only when m = 1 or n-1. When 1 &lt; m &lt; n-1, we can only prove a weaker statement, namely that there is a coloured graph C which is the universal m-cover.
The Geometry of Four-Manifolds
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
The last ten years have seen rapid advances in the understanding of differentiable four-manifolds not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics and have proved to be a mainspring of current mathematical research. This book provides a lucid and accessible account to the modern study of the geometry of four-manifolds. Consequently, it will form required reading for all those mathematicians and theoretical physicists whose research touches on this topic. Prerequisites are a firm grounding in differential geometry and topology as might be gained from the first year of a graduate course. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology - notably the definition of new invariants of four-manifolds - and also a wide-ranging treatment of relevant topics from geometry and global analysis. All of the main theorems about Yang-Mills instantons on four-manifolds are proved in detail. On the geometric side, the book contains a new proof of the classification of instantons on the four-sphere, together with an extensive discussion of the differential geometry of holomorphic vector bundles. At the end of the book the different strands of the theory are brought together in the proofs of results which settle long-standing problems in four-manifolds topology and which are close to the frontiers of current research.
Design-Theoretic Fundamentals
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
Combinatorial designs appear in a wide variety of disguises, and with a number of different restrictions. Generically, a combinatorial design consists of a ground set of elements and a collection of subsets of these elements. What distinguishes most combinatorial designs from an arbitrary collection of subsets is the requirement for balanceor regularityin some regard. For example, equality in the sizes of the subsets, or the number of subsets in which each t-subset of elements occurs, or the sizes of the intersection of pairs of subsets, may be required. In our investigation, the central restriction is that the design is t-balanced:every unordered set of telements is required to appear in the same number, λ of subsets in the collection. We are particularly concerned with designs that are 2-balanced.
Subsystems and Holes
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
One method to produce incomplete triple systems is to remove all triples of a subsystem to form the hole. However, only those incomplete triple systems in which the size of the hole is &gt;.-admissible can be produced in this way. Hence many results originally proved for triple systems with subsystems have been generalized to triple systems with holes.
Techniques of Semigroup Theory
Authors: Higgins Peter M 
Published: November 2023

Abstract
This book introduces recently developed ideas and techniques in semigroup theory to provide a handy reference guide previously unavailable in a single volume. The opening chapter provides sufficient background to enable the reader to follow any of the subsequent chapters, and would by itself be suitable for a first course in semigroup theory. The second chapter gives an account of free inverse semigroups leading to proofs of the McAlister P-theorems. Subsequent chapters have the underlying theme of diagrams and mappings, and the new material includes the theory of biordered sets of Nambooripad and Easdown, the semigroup diagrams of Remmers and Jackson with applications to the one-relator, and other word problems, a short proof of Isbell's Zigzag theorem with applications to epimorphisms and amalgams, together with combinatorial, probabalistic and graphical techniques used to prove results including Schein's Covering Theorem and Howie's Gravity Formula for finite full transformation semigroups. Nearly two hundred exercises serve the dual purpose of illustrating the richness of the subject while allowing the reader to come to grips with the material.
Free inverse semigroups and the theorems of McAlister
Authors: Higgins Peter M 
Published: November 2023

Abstract
The existence of the free inverse semigroup FI x was established in Chapter 1 (Theorem 1.1.10). A canonical form for the elements of Fix is afforded by their representation as the birooted word trees of Munn (1974). This in tum allows us to construct an alternative representation of the members ofF Ix due to Scheiblich (1972, 1973). Consideration of the semilattice ofFix leads to a further description ofFix as a so-called P-semigroup. This motivates the study of arbitrary P-semigroups which are then shown to comprise a class of £-unitary inverse semigroups. The famous theorem of McAlister (1974b) is that the converse is true: any £-unitary inverse semigroup is isomorphic to a P-semigroup. In proving this theorem we follow Munn (1976). Finally, we prove that any inverse semigroup is the image under an idempotent-separating homomorphism of an £-unitary semigroup. Again we do not record the original proof of McAlister (1974a), but establish the result by using a semidirect product construction of Wilkinson (1983), although the result can also be quickly obtained from the knowledge gained concerning Fix (Exercise 2.2.3).
Cohomology of wreath products
Authors: Evens Leonard 
Published: November 2023

Abstract
Wreath products (defined below) seem to be ubiquitous in group theory and its applications. They appear implicitly in the work of 19th century group theorists, but were first defined explicitly by Polya when studying combinatorial problems associated with classifying organic molecules (see Polya and Read 1987, p.99). They were studied systematically in a series of papers by Kaloujnine and Krasner (1948, 1950, 1951a, 1951b). The Sylow subgroups of symmetric groups (Hall 1959, Section 5.9) are formed from wreath products, as are the Sylow subgroups of Gl(n, F q) (and other linear groups) away from the characteristic of the field (Weir 1955). The cohomology of wreath products has played an important role in the development of the subject, in part because of interest in the cohomology of such groups, but also because wreath products are closely connected with the notion of transfer (corestriction) and its generalizations. We shall explore these notions in what follows.
The polynomial 3-primes generating function
Authors: Effinger Gove W , Hayes David R 
Published: November 2023

Abstract
In this and the next chapter, we apply the concepts and results of Chap ters 2 to 5 to obtain an asymptotic solution to the polynomial 3-primes problem. Throughout this chapter, a summation over ‘P’ indicates a sum mation over positive, irreducible polynomials. We begin with a brief look at what lies ahead.
Spectra
Authors: Yves Diers 
Published: November 2023

Abstract
The prime spectrum of an object A is the set of prime congruences on A with a topology called the Zariski topology. The open sets are defined by congruences on A, the closed sets by regular quotients of A, the affine open sets by codisjunctable congruences on A or singular quotients of A, and the affine subsets by simultaneously codisjunctable families of congruences on A or semisingular quotients of A. The maximal spectrum of A is the subspace of the prime spectrum whose elements are the maximal congruences on A. The patch spectrum of A is the patch space of the prime spectrum of A or the co-universal boolean space associated to it. Many properties of objects and morphisms in Zariski categories have nice topological interpretations via their spectra. In particular, the study of direct factors of objects, and of indecomposable objects, is considered here. The prime spectrum of an object A is canonically equipped with a sheaf of local objects whose object of global sections is precisely A. This is the structure sheaf on the prime spectrum. Many properties of objects and morphisms in Zariski categories can then be localized via the stalks of the structure sheaf. Indeed, this is because in a Zariski category the class of local objects is a strong cogenerating class.
Zariski Toposes
Authors: Yves Diers 
Published: November 2023

Abstract
The category of schemes on a Zariski category can be fully and left-exactly embedded in a large topos - its large Zariski topos -while the category of locally finitely presentable schemes can be fully and left-exactly embedded in a topos-its Zariski topos. Modelled toposes and locally modelled toposes can be defined relatively to a Zariski category. They efficiently play the role of ringed toposes and locally ringed toposes. The Zariski topos is the classifying topos for locally modelled toposes, and to any modelled topos is associated a universal locally modelled topos- its prime spectrum. In order to describe the relations existing between the different kinds of schemes, modelled spaces, modelled toposes, and Zariski toposes associated to different Zariski categories, it is time to introduce the notion of morphisms of Zariski categories. There are, in fact three notions: morphisms, geometrical morphisms, and cogeometrical morphisms.
Some Constructions Of Zariski Categories
Authors: Yves Diers 
Published: November 2023

Abstract
Some general constructions of categories can be used to build up Zariski categories starting with a Zariski category A. The coslice construction is extensively used: for any fixed object A in A, the coslice category A/ A of objects of A under A is a Zariski category. The functor category construc tion can also be used: if C is a small category with a strict initial object, the functor category Ac is a Zariski category. In particular, the category A → of morphisms of A is a Zariski category. Finite products of Zariski categories are Zariski categories. Some categories of sheaves are Zariski categories: if Xis a boolean topological space the category Sh [X, A] of sheaves on Xwith values in A is a Zariski category. Some full subcategories of A are Zariski categories, as the subcategories of reduced, regular, neatish, or etalish objects of A.
On maximal sets of nuclei in PG(2,. q) and quasi-odd sets in AG(2, q)
Authors: Blokhuis A , Mazzocca F 
Published: November 2023

Abstract
The proof of Result 1 surprisingly does not use Result 2. In this note we start with an elementary derivation of 1 from 2. This is done by a process we call “lifting”. The following paragraph discusses “quasi-odd” sets and we proceed by establishing the connection between sets with the maximal number of nuclei and quasi-odd sets. We finish with odds and ends.
On an exceptional semi-translation plane
Authors: Resmini M J de 
Published: November 2023

Abstract
In 1970 N.L. Johnson (1970) constructed an exceptional semi translation plane of order 16, say 1t, which is of Lenz-Barlotti class 1-Sa and can be derived from a dual translation plane, namely the semifield plane of order 16 with kern GF(4).
Descent by 3-isogeny and 3-rank of quadratic fields
Authors: Top Jaap 
Published: November 2023

Abstract
In this paper families of elliptic curves admitting a rational isogeny of degree 3 are studied. It is known that the 3-torsion in the class group of the field defined by the points in the kernel of such an isogeny is related to the rank of the elliptic curve. Families in which almost all the curves have rank at least 3 are constructed. In some cases this provides lower bounds for the number of quadratic fields which have a class number divisible by 3.
Lattice-based models
Authors: Grenander Ulf 
Published: November 2023

Abstract
We are now ready to apply the general principles of Chapters 13–14, and start by doing this for the lattice-based models introduced in Section 8.2.A simple but important case is when D is just a random generator map meaning that 𝒟 takes g into g′ where g′ has a given conditional probability distribution for fixed g; all gi treated independently.
Creating generators
Authors: Grenander Ulf 
Published: November 2023

Abstract
The hardest task in pattern theory is the invention of regular structures and their components: generators with bonds, bond relations, acceptor functions, and so on. This requires real insight into the logical structure of the patterns under study, and little constructive advice can be given in general: usually each case has to be treated on its own. We get some help from the morphogenetic devices in Section 4.3. There are, however, some cases where it is possible to offer concrete suggestions for some of these tasks, and we shall do this in the present and next chapters.
Lie polynomials
Authors: Reutenauer Christophe 
Published: November 2023

Abstract
After introducing words, noncommutative polynomials and series, we define Lie polynomials. One of the main results presents the various characterizations of Lie polynomials. This naturally leads us to define the shuffle product, and to study the duality between concatenation and shuffle product; in other words: the Hopf-algebra-like properties of the free associative algebra. Related questions are treated in the appendix, including the support of the free Lie algebra (the set of words which may appear in Lie polynomials), the free Lie p-algcbra and the Jacobson identities, the kernel of the left to right bracketing mapping and a brief excursion into automata theory.
The Solomon descent algebra
Authors: Reutenauer Christophe 
Published: November 2023

Abstract
The convolution subalgebra generated by the projections of the free associative algebra arising from its graduation is also an algebra under composition, and is canonically isomorphic with a subalgebra of the symmetric group algebra, called the descent algebra; the latter was introduced by Solomon for each finite Coxeter group. The canonical projections of the free associative algebra, arising from its structure of enveloping algebra of the free Lie algebra, correspond to the primitive idempotents of the descent algebra; this is established in Section 9.2, where some insight into the structure of this algebra is also given. In the next section, we study various homomorphisms of this algebra: one is a mapping into the ring of symmetric functions, and its kernel is the radical of the descent algebra; another maps each descent algebra into another one, and is a derivation with respect to the convolution product. Elements of the descent algebra are also characterized, in the symmetric group algebra, by their action on Lie monomials. In the final section, we introduce quasisymmetric functions, which are closely related to the descent algebra, and we give an application to the enumeration of permutations.