bluuebunny/crossref_metadata_2025 huggingface dataset:
Project 5
Authors: Grandine Thomas A
Published: November 2023

ACME has a problem. Wile E. Coyote, one of ACME’s best clients for many years, is having trouble with his Locational Operative Responsive Avian Network unit, built by ACME to use radar to track down desert fowl. The LORAN unit provides enough information for most applications, but not for the application which Wile E. Coyote has in mind. The target he would most like to watch moves more quickly than the LORAN unit was designed to handle. Thus, he needs roughly ten times as much data transmitted in any given time period as he is currently getting. Because ACME needs to keep Wile E. Coyote as a customer, and because the management would rather not shell out the money to design a much improved LORAN unit, you have suggested to your boss that interpolation might solve the problem. In particular, you have agreed to help Wile E. Coyote obtain the necessary data to catch the desert bird he is after. If you are successful, ACME will retain Wile E. Coyote’s business. If you fail, then he will almost certainly transfer his business to ACME’s chief competitor, AJAX.
Project 8
Authors: Grandine Thomas A
Published: November 2023

Based on your past successes, Biro Kratt has recommended that you assist on a project of great importance to ACME. The Republic United Steel Truss division of ACME manufactured steel beams for a customer, Monolithic Engineering Steel Structures. These beams were used in the construction of a structure called the Unduly Gigantic Hanger. Several of the UGHs have shown signs of severe fatigue, and MESS is suing ACME, claiming that the beams produced by RUST are at fault. RUST maintains that the beams were made to specifications and believes that UGH is so constructed that the beams are subjected to forces beyond specifications. Your task is to analyze UGH to determine the forces to which the beams are subjected. The UGH is constructed with twenty seven (27) beams, or members, connected at eleven (11) joints. MESS has specified that each beam is required to withstand extension forces of up to 20 tons and compressive forces up to 17 tons without fatigue failure. Through a court order, ACME has obtained from MESS the data on the UGH. The figure shows what the UGH looks like. The loads are attached to joint 11.
Project 11
Authors: Grandine Thomas A
Published: November 2023

The Mechanical Operations Utility Staff for Engineering Technology for Research And Production, a division of ACME, is currently designing a new motor to be used in many of its products. The old motor is large, heavy, and relatively energy inefficient by modern standards, so ACME has decided to invest in the development of an improved motor to improve the quality of the products. Thus, many of the division’s employees are currently working on this project in order to build a better MOUSETRAP motor. In any case, the new MOUSETRAP motor project is well underway. In fact, a prototype is already functioning in the lab. However, the testing of this new motor is far from complete. One of the current projects is to determine the startup response of the motor. The MOUSETRAP lab has already made several measurements, but they do not yet have a good mathematical model of what is going on. They have asked SOLVE for assistance, and this assignment has landed in your lap. Being the helpful individual that you are, you have happily agreed to solve this problem.
Energy
Authors: Lunn Mary
Published: November 2023

When a ski-jumper starts his run down the hill he has very little initial velocity. As he loses height so he gains speed. Conversely in order to gain sufficient height to reach a fire in a tall building the nozzle velocity of water from a fireman’s hose has to be of a high order of magnitude. There is some balancing mechanism between the work done overcoming the force of gravity and the change in the velocity of the particle. Some readers will be familiar with the one-dimensional formalism of energy. Newton’s second law leads us to a very clear description which we recapitulate here.
Rigid bodies: equations
Authors: Lunn Mary
Published: November 2023

In Newtonian mechanics a rigid body (or solid object) has a particular significance. The equations are developed from an assumption which defines the relationship between the particles which collectively form the body. A rigid body is defined to be a collection of particles such that each particle remains in the same position relative to all other particles of the body. This ensures that the body retains its size, shape, and distribution of mass.
Two Topics in Nonlinear Stability
Authors: Sanz-Serna J M
Published: November 2023

There are several ways in which the word stability may be understood in a numerical analysis context. First of all, stability is used in expressions like ‘stability and consistency imply convergence’. In this first sense, stability refers to dependence of a numerical result on the data, and, as such, applies to most numerical computations, including numerical linear algebra, quadrature, differential equations, etc. This first notion is akin to the idea of ‘well posedness’. In fact, in many cases, a numerical procedure is said to be stable in this sense if it is well posed uniformly with respect to the relevant parameters, such as the dimension of the problem, grid-size, etc. In a second use, the term stability applies in connection with the long time behaviour of discretizations of time-dependent problems in ordinary or partial differential equations. The stability of discretizations of nonlinear differential equations, ordinary or partial, is the unifying theme of the present work. In Sections 4.2 to 4.8, we deal with the question of how best to define ‘stability’ of a nonlinear discretization so that the familiar ‘stability and consistency imply convergence’ holds.
Subdivision Schemes in Computer-Aided Geometric Design
Authors: Dyn N
Published: November 2023

Computer Aided Geometric Design (CAGD) is a branch of applied mathematics concerned with algorithms for the design of smooth curves and surfaces and for their efficient mathematical representation. The representation is used for the computation of the curves and surfaces, as well as geometrical quantities of importance such as curvatures, intersection curves between two surfaces and offset surfaces.
The Theory of Radial Basis Function Approximation in 1990
Authors: Powell M J D
Published: November 2023

Throughout this work {f(x) : x ∈ ℝ} is a real valued function of d variables that is to be approximated by {s(x): x ∈ ℝ}. The radial basis function method gives many useful choices of s that have several advantages over piecewise polynomials. This approach to multivariable approximation is relatively new, however, so many of the most exciting properties have been proved to hold only in special cases. A coherent account of the known theory is presented. It includes some stunning results on orders of convergence and guaranteed nonsingularity of interpolation matrices. Therefore the author believes that radial basis function methods are becoming as important as piecewise polynomials although good software is not yet available for general computer calculations. We shall enjoy the theoretical discoveries that have been made so far and we will note some interesting conjectures. Earlier reviews of theory are presented by Dyn [12,13]. Many successful applications of radial basis functions are mentioned in ·a recent paper that provides an excellent survey of the practical side of the subject (Hardy [22]).
Smoothing Spline Curve Fitting
Authors: Dierckx Paul
Published: November 2023

In Chapter 4 we saw that we can obtain excellent approximation results with free knot least-squares splines. A serious drawback of the method is its computational cost which increases enormously with the number of knots. Moreover, as this number increases, the chance of missing the optimal location, because of an inappropriate initial position, also grows. Therefore, the method should not be recommended for data sets for which it is not possible to obtain a satisfactory fit with, say, fewer than 10 interior knots.

In this chapter we will consider another approximation criterion which is specific to the use of splines. It will allow us to determine a satisfactory fit in a much faster and more flexible way. The knots will again be located automatically but here the knot placing strategy will only aim to be reasonably frugal with knots, not to find the minimal number nor the optimal location.
Scattered Data Surface Fitting
Authors: Dierckx Paul
Published: November 2023

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

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

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

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.
Performance of Lapack
Authors: Dongarra Jack
Published: November 2023

Can we provide portable software for computations in dense linear algebra that is efficient on a wide range of modern high-performance computers? If so, how? Answering these questions — and providing the desired software — has been the goal of the LAPACK project.
An application to number theory: prime integer points
Authors: Huxley M N
Published: November 2023

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

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

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

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

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

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

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

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

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.
Intersection Properties
Authors: Pasini Antonio
Published: November 2023

In this chapter Γ is a geometry of rank n over a set of types I with type function t, and J is a nonempty subset of I. According to the convention stated in Section 4.4, the geometry Γadmits a diagram.
Universal Covers
Authors: Pasini Antonio
Published: November 2023

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 < m < 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

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

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

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 >.-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

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

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

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

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

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

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

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

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

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).
Strongly regular graphs induced by polarities of symmetric designs
Authors: Haemers W H , Higman D G , Hobar S A
Published: November 2023

In the present paper, we study strongly regular polar decompositions. It is, in a certain sense, an extension to a paper on strongly regular graphs with strongly regular decompositions by Haemers and Higman (1989).
Solutions of quadratic equations in a groupring and partial addition sets
Authors: Löwe S
Published: November 2023

Proof. By the assumption of the theorem, both Δ and Θ are normal subsets of G. From the definition it is clear that both are invariant under taking the inverse.
Common characterizations of the finite Moufang polygons
Authors: State H Van Maldeghem
Published: November 2023

The notion of a generalised polygon was introduced by Tits (1959). The main idea was to give a geometric interpretation of the Chevalley groups and the twisted groups of rank 2.
Descent by 3-isogeny and 3-rank of quadratic fields
Authors: Top Jaap
Published: November 2023

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.
Generators and configurations
Authors: Grenander Ulf
Published: November 2023

General pattern theory starts from a few simple but fundamental principles that will guide us when building regular structures to represent knowledge about patterns.
Lattice-based models
Authors: Grenander Ulf
Published: November 2023

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

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

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

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.
Constructive Approximation on the Sphere
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

The subject of geomathematics focuses on the interpretation and classification of data from geoscientific and satellite sources, reducing information to a comprehensible form and allowing the testing of concepts. Sphere oriented mathematics plays an important part in this study and this book provides the necessary foundation for graduate students and researchers interested in any of the diverse topics of constructive approximation in this area. This book bridges the existing gap between monographs on special functions of mathematical physics and constructive approximation in Euclidean spaces. The primary objective is to provide readers with an understanding of aspects of approximation by spherical harmonics, such as spherical splines and wavelets, as well as indicating future directions of research. Scalar, vectorial, and tensorial methods are each considered in turn. The concentration on spherical splines and wavelets allows a double simplification; not only is the number of independent variables reduced resulting in a lower dimensional problem, but also radial basis function techniques become applicable. When applied to geomathematics this leads to new structures and methods by which sophisticated measurements and observations can be handled more efficiently, thus reducing time and costs.
Notation
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

For the convenience of the reader we provide in this introductory chapter a summary of mathematical notation that will be used throughout the scalar part of the book. We begin with the customary set-theoretical preliminaries and some basic settings in Euclidean spaces.
Spherical Singular Integrals
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

We now study some important facts about spherical singular integrals. Since most of the material about singular integrals is rather standard our treatment will be brief. Three types of singular integrals on the sphere will be discussed in more detail, which are of basic interest in numerical applications.
Algorithms for Boolean Formula Evaluation and for Tree Contraction
Authors: Buss* Samuel R
Published: November 2023

Abstract

We present a new, simpler Alogtime algorithm for the Boolean sentence value problem (13SVP). Unlike prior work, this algorithm avoids the use of postfix-longer-operand-first formulas. We also show that tree-contraction can be made Alogtime uniform.

The Boolean sentence problem restricted to balanced sentences with only the connectives ⋀ and ⋁ is in online O(log(log n)) space. Hence every Alogtime predicate is deterministic log-time reducible to log(log n) space. This balanced and/or BSVP has logarithmic width, linear length, branching programs.
Stochastic geometry
Authors: Kingman J F C
Published: November 2023

So far we have considered random countable subsets of spaces S which have always been subsets of !Rd for some dimension d. The theory is, however, much more general, and applies to random structures which look quite different from the irregular array of isolated points depicted in Fig. 1.1. Suppose for instance that one looks through a microscope at a piece of paper. The view might roughly resemble Fig. 7.1, where the lines represent fi bres of the material. This could be modelled by representing the fibres as infinite straight lines, or in a more complex way allowing for thickness, finite length or departure from straightness. The lines could be regarded as lying in a plane, or in three dimensions if the thickness of the material justified it.
Evolving Curves
Authors: Gurtin Morton E
Published: November 2023

This chapter discusses the kinematics of curves that evolve smoothly in time, and forms the basis of our discussion of phase interfaces.

Lett denote the time.
Constitutive Theory
Authors: Gurtin Morton E
Published: November 2023

Constitutive equations and the compatibility theorem As constitutive equations we allow the energy, the capillary force, and the normal interaction to depend smoothly on the orientation of the interface through a dependence on θ and on the kinetics of the interface through a dependence on
Thermodynamics of Two-Phase Systems
Authors: Gurtin Morton E
Published: November 2023

We consider an evolving interface, 𝓈 separating two phases, labelled 1 and 2, and write Ω1  (t) and Ω2(t) for the corresponding phase regions as discussed in Section 3.1. The normal N for, 𝓈 then coincides with the outward unit normal to ∂ Ω1. For themoment we will restrict attention to situations in which the body Ω = Ω1(t) ∪ Ω2(t) is all of ℝ2.
Universal Algebra
Authors: Meinke K , Tucker J V
Published: November 2023

This chapter is an introduction to the primary concepts, constructions and results about algebras, their isomorphism and axiomatization. As with other chapters in this Handbook, it is composed with the aims of teaching the mathematical theory and explaining its relevance to problems in computer science. Among areas of application we will consider are: abstract data types and their specification; syntax and semantics of programming languages; and synchronous concurrent algorithms and architectures. Some of these algebraic applications, and others such as asynchronous concurrent processes, are developed fully as the subject matter of chapters in later volumes of the Handbook.
Mathematical Induction
Authors: Walther Christoph
Published: November 2023

Proving statements of arithmetic usually requires induction. Statements like are well-known from basic mathematics courses and provide beginner’s exercises for applying the induction principle. If we prove such statements we often appeal to our ‘intuition’ and use our knowledge about arithmetic without being aware of each decision made during the search for a proof. Of course, this is not required as long as we are successful in proving theorems by induction. But if we intend to automate induction theorem proving our proof-strategic knowledge must be made explicit such that a computer can perform the right decisions when searching for a proof.
Higher Order Logic
Authors: Leivant Daniel
Published: November 2023

Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foundational aspects of higher order logic, and also to briefly point to some areas of current and potential applications. The chapter falls into two parts. The first part, consisting of sections 1 through 4, is designed to bring forth the essential issues and facts of the topic. Section 1 is intended to motivate our interest in higher order logic, by presenting selected examples of the expressive power of second order logic, and contrasting them with the limitations of first order logic.
Defeasible Logic
Authors: Nute Donald
Published: November 2023

Like all reasoning, defeasible reasoning involves inference. One makes an inference when one comes to have some belief on account of other beliefs that one has. The kind of relationship I intend by ‘on account of’ is one in which a person who makes an inference does or would put forward the beliefs on account of which she comes to have the new belief as reasons for accepting the new belief. This is very rough, but my main goal is not to explain inference. What I am concerned to do is to determine how the inferences made in defeasible reasoning, which I will call defeasible inferences, are different from other inferences..
Possibilistic Logic
Authors: Dubois D , Lang J , Prade H
Published: November 2023

Possibilistic logic is a logic of uncertainty tailored for reasoning under in-complete evidence and partially inconsistent knowledge. At the syntacticlevel it handles formulae of propositional or first-order logic to which are attached numbers between 0 and 1, or, more generally, elements in a to-tally ordered set. These weights are lower bounds on so-called degrees of necessity or degrees of possibility of the corresponding formulae. The de-gree of necessity (or certainty) of a formula expresses to what extent the available evidence entails the truth of this formula.
Domain Theory
Authors: Abramsky Samson , Jung Achim
Published: November 2023

Let us begin with the problems which gave rise to domain theory: Least fixpoints as meanings of recursive definitions. Recursive definitions of procedures, data structures and other computational entities abound in programming languages. Indeed, recursion is the basic effective mechanism for describing infinite computational behaviour in finite terms.
Temporal Logic
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

This long awaited book gives a thorough account of the mathematical foundations of Temporal Logic, one of the most important areas of logic in computer science. The book, which consists of fifteen chapters, moves on from giving a solid introduction in semantical and axiomatic approaches to temporal logic to covering the central topics of predicate temporal logic, meta-languages, general theories of axiomatization, many dimensional systems, propositional quantifiers, expressive power, Henkin dimension, temporalization of other logics, and decidability results. Much of the research presented here is frontline in the new results and in the unifying methodology. This is an indispensable reference work for both the pure logician and the theoretical computer scientist.
The handling of time: introduction and survey
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

This book is about the handling of time, both in pure logic and in computing. It attempts to present a unified point of view, where computational and mathematical properties of the logic of time are interwoven in an intrinsic way.
Groups of exponent two, three, and six
Authors: Vaughan-Lee Michael
Published: November 2023

In this chapter we prove that groups of exponent 2, 3, and 6 are locally finite, and we obtain some information about the structure of the Burnside groups of these exponents. (A group is said to be locally finite if all its finitely generated subgroups are finite.)
Groups of prime exponent
Authors: Vaughan-Lee Michael
Published: November 2023

We let B(r, p) be the r generator Burnside group of exponent p, and we let R(r, p) = B(r, p)/K, where K is the intersection of all subgroups of finite index in B(r, p). Thus every finite r generator group of exponent p is a homomorphic image of R(r, p). Kostrikin’s solution of the restricted Burnside problem for prime exponent implies that if p is prime then R(r, p) is finite for r = 1, 2, … On the other hand, Adjan’s solution of the unrestricted Burnside problem implies that if p ≥665 then B(r, p) is infinite for r≥ 2.
Groups of prime-power exponent
Authors: Vaughan-Lee Michael
Published: November 2023

We consider groups of prime-power exponent q =pk> 4 (with k > 1). Adjan’s solution of the Burnside problem [l] tells us that if r≥2, q ≥665, and p > 2 then B(r, q) is infinite. Zelmanov’s solution of the restricted Burnside problem for groups of prime-power exponent implies that L(r, q) is nilpotent, but Razmyslov’s Theorem tells us that L(r, q) has derived length at least 1 + [log2  r]. We look at the Lie rings of groups of exponent pk in §8.1. In §8.2 and §8.3 we look at some results in exponents 8 and 9.
Random Walks And Random Environments
Authors: Hughes Barry D
Published: November 2023

This is the first volume of a two-volume work devoted to probability theory in physics, physical chemistry and engineering. This volume provides an introduction to the problem of "random walk" and its applications. In its simplest form, the random walk describes the motion of an idealized drunkard and is a discrete analogue of the diffusion through a medium with traps, laser speckle and the conformations of polymers in dilute solution. Prior knowledge of probability theory is helpful, but not assumed.
Random Walks and Random Environments
Authors: Hughes Barry D
Published: November 2023

This is the second volume of a two-volume work devoted to probability theory in physical chemistry, and engineering. Rather than dealing explicitly with the idea of an ongoing random walk, each chaotic step taking place at fixed time intervals, this volume addresses models in which the disorder is frozen in space-random environments. The volume begins with a largely self-contained introduction to the geometry of random environments, emphasizing Bernoulli percolation models. The scope of the investigation then widens as we ask how structural disorder affects the transport process. The final chapters confront the interplay of two different forms of randomness; spatial randomness frozen into the environment and temporal randomness associated with the choices for next steps made by a random walker. The book ends with a discussion of "the ant in the labyrinth" problems. It is supported by an extensive bibliography and very little prior knowledge is assumed.
Time and Change for AI
Authors: Galton Antony
Published: November 2023

Outside physics and philosophy one seldom reasons much about time and change in the abstract, but on the other hand reasoning about particular times and changes—dates, processes, events, and so on—is all-pervasive both in the specialist sciences and in everyday life. No account of reasoning can properly be called complete if it does not say something about how we reason about change; and if, as in Al, the aim is to simulate or duplicate human reasoning computationally, then the account of change required must be both extensive and detailed. In this section we survey six areas pertaining to Al in which a need for temporal reasoning has been felt, introducing some of the key issues arising in these areas; in later sections we examine these issues in greater detail.