bluuebunny/crossref_metadata_2025 huggingface dataset:
Competitive unsupervised learning processes
Authors: Coolen A C C , Kühn R , Sollich P
Published: November 2023

In this chapter, we introduce and study two so-called ‘unsupervised’ learning processes, where the problem is not to learn to associate some correct response to each possible incoming signal (specified by a teacher or supervisor signal, as in, for example, feed-forward networks of the perceptron or multilayer perceptron type), but rather to build some alternative (more efficient, more compact, or more structured) representation of the data vectors fed into the system. The first two procedures to do this, vector quantization (VQ) and soft vector quantization (SVQ), aim to achieve data reduction for initially unknown data distributions. They find typical applications in communication, where compact representations are obviously cheaper to transmit than non-compact ones, allowing more signals to be communicated via the same channel. Methods in a third class—self-organizing maps (SOM), also known as feature maps or Kohonen maps—aim to create a topologically correct but low-dimensional internal representation of a given data distribution. This has applications in, for example, biology (the brain is known to create such maps for sensory signals), data-base mining, medicaldiagnostics, etc.
Bayesian techniques in supervised learning
Authors: Coolen A C C , Kühn R , Sollich P
Published: November 2023

In this chapter, we discuss the Bayesian approach to supervised learning from examples. Its introduction in the 1990s has been a very welcome development, in that it contributed significantly to converting artificial neural computation, especially learning in multilayer networks, from a collection of interesting but ad hoc algorithms (‘invent a rule, play with parameters, draw performance graphs, and hope for the best’) to a sound, well grounded, and systematic scientific procedure. It also contributed to the area becoming more mathematical in nature.
Identification of entropy as an information measure
Authors: Coolen A C C , Kühn R , Sollich P
Published: November 2023

In this chapter, we will use coding theory to prove that the entropy does indeed represent the information content of a set of messages generated according to a set of prescribed probabilities. We will first do this by showing that the average number of bits used per message is equal to the entropy H , if we use the optimal code to represent the possible values of the messages (i.e. the one with the smallest average length of codewords). There-after, we shall also present the main line of reasoning of Shannon’s elegant original proof, which is based on a few basic assumption about properties of a sensible measure of information, and does not refer to optimal codes.
Applications to neural networks
Authors: Coolen A C C , Kühn R , Sollich P
Published: November 2023

The main contribution of information theory to the field of information processing in natural or artificial neural networks is that it provides exact performance measures. This allows us to compare the performance of different models or algorithms in a rigorous way. It also lets us develop learning rules based on the maximization of these information-theoretic performance measures, which are no longer ad hoc and also apply in unsupervised scenarios. The examples we will discuss here are the Boltzmann machine learning rule for recurrent layered neural networks and various types of learning rules (e.g. maximum information preservation) for layered networks. In addition we will touch on the topic of learning by natural gradient descent, based on thinking in terms of distances between probability distributions (i.e. information geometry).
Network operation: macroscopic dynamics
Authors: Coolen A C C , Kühn R , Sollich P
Published: November 2023

In non-equilibrium statistical mechanics one aims to derive from the laws of the underlying microscopic system x, and solve, dynamical equations for a suitable small set of macroscopic quantities Ω (x) = (Ω1(x), ... , Ωn(x)). In principle this can be done in two ways. The first route (clockwise in the diagram), which is normally too complicated, would consist of solving the microscopic stochastic equations directly. From the solution one then calculates the values of the macroscopic quantities. The second route (anticlockwise) consists of first calculating from the microscopic stochastic laws a dynamical equation for the macroscopic probability distribution Pt (Q), which then has to be solved. If applicable, this is the easiest and preferred approach.
Dynamics of online gradient descent learning
Authors: Coolen A C C , Kühn R , Sollich P
Published: November 2023

In this chapter we consider learning in systems with continuous real-valued outputs, rather than those with binary outputs as in the previous chapter. A natural way of constructing learning algorithms for such systems is by gradient descent on some appropriate error measure. This error measure should tell us by how much student and teacher output differ. A popular choice is the squared deviation between actual and desired outputs; carrying out gradient descent on this measure for multilayer networks yields the well-known error backpropagation algorithm. The results of our analysis are therefore of relevance to practical neural network learning.
Network operation: equilibrium analysis
Authors: Coolen A C C , Kühn R , Sollich P
Published: November 2023

In this chapter we restrict ourselves to networks which obey detailed balance, so that we know the equilibrium probability distribution and equilibrium statistical mechanics applies. We will only consider the case of sequential dynamics; the results for parallel dynamics can be derived along very similar lines, and are largely identical.
Imaging spatio-temporal patterns of long-term potentiation in mouse hippocampus
Authors: Hosokawa Toshiyuki , Ohta Masaki , Saito Takeshi , Fine Alan
Published: November 2023

Spatio-temporal patterns of neuronal activity before and after the induction of long-term potentiation in mouse hippocampal slices were studied using a real-time high-resolution optical recording system. After staining the slices with voltage-sensitive dye, transmitted light images and extracellular field potentials were recorded in response to stimuli applied to CAI stratum radiatum. Optical and electrical signals in response to single test pulses were enhanced for at least 30 minutes after brief high-frequency stimulation at the same site. In two-pathway experiments, potentiation.
Cadherins and synaptic plasticity: activity-dependent cyclin-dependent kinase 5 regulation of synaptic-catenin-cadherin interactions
Authors: Schuman Erin M , Murase Sachiko
Published: November 2023

Cyclin-dependent kinase 5 (Cdk5)/p35 kinase activity is known to decrease the affinity of P-catenin for cadherin in developing cortical neurons. Our recent work demonstrated that depolarization causes an increased affinity between P-catenin and cadherin. Here, we examine whether Cdk5/p35 regulates P-catenin-cadherin affinity in response to neural activity. In hippocampal neurons depolarization caused a significant decrease in Cdk5 kinase activity, without changing the protein levels of either Cdk5 or p35, suggesting that the proteasome pathway is not involved. Decreasing Cdk5 kinase activity with the inhibitor roscovitine increased the amount of P-catenin that was co-immunoprecipitated with cadherin. Inhibiting Cdk5 activity also resulted in a redistribution ofEGFP-P-catenin from the dendritic shaft to the spines, where cadherins are highly concentrated.
Inducible molecular switches for the study of long-term potentiation
Authors: Hedou Gael , Mansuy Isabelle M
Published: November 2023

This article reviews technical and conceptual advances in unravelling the molecular bases of long-term potentiation (LTP), learning and memory using genetic approaches. We focus on studies aimed at testing a model suggesting that protein kinases and protein phosphatases balance each other to control synaptic strength and plasticity. We describe how gene ’knock-out’ technology was initially exploited to disrupt the Ca2+ /calmodulin-dependent protein kinase Ilc.t (CaMKifo) gene and how refined knock-in techniques later allowed an analysis of the role of distinct phosphorylation sites in CaMKII. Further to gene recombination, regulated gene expression using the tetracycline-controlled transactivator and reverse tetracycline-controlled transactivator systems, a powerful new means for modulating the activity of specific molecules, has been applied to CaMKlfo and the opposing protein phosphatase calcineurin.
Quantum Groups
Authors: Etingof Pavel , Latour Frédéric
Published: November 2023

The material in this chapter is standard. It can be found in textbooks on quantum groups, such as Kassel (1995), Etingof and Schiffmann (2002), Jantzen (1996), Lusztig (1993).
Dynamical Weyl Group
Authors: Etingof Pavel , Latour Frédéric
Published: November 2023

In this chapter we will review the theory of the dynamical Weyl group. This theory has many applications, in particular to study of traces of intertwining operators. The theory of the dynamical Weyl group was developed in Tarasov and Varchenko (2000), Etingof and Varchenko (2002).
The stars on the astrolabe
Authors: Kunitzsch Paul
Published: November 2023

The Greek word ‘astrolabe’ means ‘star catcher’. This name exactly describes an essential element of the practical use of the instrument. By means of a sighting device – called the ‘alidade’ (Arabic al-cicada ) – which is placed on the back of the instrument, one can measure the altitude of a heavenly body above the horizon. Then, the position of a star, the Sun, or the Moon can be set by turning the ‘rete’ (‘net’ or ‘spider’; Arabic al-cankabut , Latin alhancabuth , etc.) located on the front of the instrument. The rete is a perforated plate which represents a two-dimensional map of the heavens. The main structural elements of the rete include the ring of the ecliptic and one or more supporting bars which are used to connect the elliptical ring to the circumference of the plate. A number of pointers are attached to these structural elements.
Modal Logic And Closure Algebras
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

The reader who has no previous familiarity with modal logic may wish to consult Hughes and Cresswell (1968, 1996). The classic, but now dated reference, is Lewis and Langford (1932). The modal logics which we shall consider are only a few among many. The reader wanting to know more about the many should consult Chellas (1980), Segerberg (1971) or Gabbay (1976). Incidentally, this section is very much influenced by Lemmon (1966) and Lemmon with Scott (1977).
Representations And Duality
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

The study of representations of lattices and related structures has been considerably enriched since Stone ‘s original work, with contributions from various authors. We restrict ourselves here to a brief report for the sake of the interested reader. We begin by bringing attention to an important aspect of the Stone representation, called “duality, “ that we did not previously discuss. Let us first try to give philosophical meaning to duality issues. Duality gets its ultimate expression in the terms of category theory, but we shall do our best to give the reader some intuitive grasp of duality without resorting to category theory, or at least its diagrams.
Algebraization
Authors: Kawakubo Katsuo
Published: November 2023

We have studied transformation groups in a geometric way. In this chapter we change entirely the method employed so far and consider, so to speak, algebraization of geometric structures. Specifically, we shall study the algebraic aspects of transformation groups. The essence of transformation groups will be abstracted, recognized, and understood via algebraization. As a means of algebraization, we introduce the concept of G-complexes for G, a finite group, and study first the relationships among the Euler characteristics of Gcomplexes, orbit spaces, and various kinds of fixed point sets. The Mobius inversion formula plays an important role in this study. When G is a compact Lie group, similar formulae for Euler characteristics of G∼manifolds are also obtained by making use of the decomposition theorem in Section 4.6 (see Section 5.6).
Trees
Authors: Read Ronald C , Wilson Robin J
Published: November 2023

A tree is a connected graph that has no cycles. If a tree has n vertices, then it has n − 1 edges, and any tree with more than one vertex has at least two end-vertices (vertices of degree 1). In any tree, there is a unique path between any two given vertices, and the removal of any edge disconnects the tree.

A rooted tree is a tree in which a particular vertex, the root, has been singled out. A tree is homeomorphically irreducible if it has no vertices of degree 2. An identity tree is a tree whose only automorphism is the identity map—that is, a tree with no non-trivial symmetries. For example, tree (a) below is homeomorphically irreducible, and tree (b) is an identity tree. A binary tree is a rooted tree in which at each vertex there can be a left upward branch or a right upward branch, or both or neither (see tree (c)).
Regular Graphs
Authors: Read Ronald C , Wilson Robin J
Published: November 2023

A graph is regular if all its vertices have the same degree; in particular, graphs that are regular of degree 3, 4, 5 and 6 are called, respectively, cubic, quartic, quintic and sextic. An example of each type is shown below.

A regular graph is polyhedral if its vertices and edges correspond to the vertices and edges of a 3-dimensional polyhedron-that is, it is planar and 3-connected. A bicubic graph is a graph that is both bipartite and cubic. The following polyhedral graphs are both cubic, but only the first is bicubic.
Types Of Graph
Authors: Read Ronald C , Wilson Robin J
Published: November 2023

Recall that a graph G is bipartite if its vertex set V can be partitioned into two sets X and Y so that each edge joins a vertex in X and a vertex in Y, and that G is Hamiltonian if it has a cycle that includes each vertex exactly once. G is an even graph if each vertex has even degree; thus, a connected even graph is an Eulerian graph. G is self-complementary if G is isomorphic to 1”5 complement; the number of vertices in a self-complementary graph must be of the form 4r or 4r + 1, where r is an integer. G is triangle-free if it has no cycles of length 3, and unicyclic if G contains only one cycle.
Signed Graphs
Authors: Read Ronald C , Wilson Robin J
Published: November 2023

A cycle in a signed graph is a positive cycle if the number of negative edges is even and is a negative cycle if the number of negative edges is odd. A signed graph is balanced if and only if each cycle is a positive cycle, and unbalanced otherwise; equivalently, a signed graph is balanced if we can colour each vertex red or green in such a way that positive edges have ends of the same colour and negative edges have ends of different colours. Signed graphs are used in the social sciences, with positive edges corresponding to positive relationships and negative edges corresponding to negative ones. Balanced signed graphs correspond to social situations in which there is no ‘tension’.
Ramsey Numbers
Authors: Read Ronald C , Wilson Robin J
Published: November 2023

The (diagonal) Ramsey number r(G) of a single graph G is the Ramsey number r(G, G). On pages 374—379 we depict all isolate-free graphs with up to 7 edges, with their Ramsey numbers. The R-number appearing in each top-right comer is the serial number of the graph in Burr’s catalogue; a graph that appears elsewhere in this Atlas is cross-referenced in the top- left corner. On page 380 we depict the graphs with more than 7 edges for which the Ramsey number is known.
A selection of elementary problems treated by eminent twentieth-century mathematicians
Authors: Gofman Judita
Published: November 2023

What kinds of research work are professional mathematicians concerned with nowadays? Answers to this question are beyond the scope of our book. Nevertheless, we shall be able to discuss here a variety of elementary, beautiful and intriguing problems which have caught the attention of eminent contemporary scholars. In addition to being intrinsically interesting, the problems chosen for this chapter are connected with important branches of modern mathematics.
Project 4
Authors: Grandine Thomas A
Published: November 2023

Because of his newly found confidence in your ability, Biro Kratt has just assigned you to a more challenging project. It appears that the ACME Super Power Usefully Transmitting Nice Information computer satellite has begun to malfunction. Your boss swears that it’s the last piece of high-tech equipment he will ever buy at a rummage sale. However, the malfunction may still be dealt with, and you have been assigned the task of overcoming the difficulty. The nature of the malfunction is that 80 per cent of the signal is getting lost in static. Your job is to try to reconstruct the signal based on the 20 per cent of the signal which is getting through. You will attempt to do this using interpolation.
Project 7
Authors: Grandine Thomas A
Published: November 2023

The Corporate Headquarters for Extraneous Accounting Procedures, an administrative group within ACME, needs to make an important business decision. CHEAP wants to borrow $100 million in order to finance an important research project whose ultimate success will prove extremely beneficial for ACME. CHEAP has located two potential sources for this money. The first, Mutual United Stock and Corporate Liabilities Exchange, is willing to loan the $100 million at an annual interest rate which has not yet been negotiated. CHEAP would be expected to repay the loan over 10 years by making ten equal payments to MUSCLE at the end of each year. The other source of money is the Security Home Annuity Refund Kartel.
Introduction: logic and language
Authors: Potter Michael D
Published: November 2023

In principle the exposition of a first-order mathematical theory need consist of no more than a list of assumptions, called axioms, followed by a list of assertions, called ,*theorems each of which is written in the language of the theory and follows from the axioms and the previous theorems by means of the rules of classical logic. Checking such a list of theorems for deductive correctness would be a very boring but entirely mechanical process and it would be easy to program a computer to do it. However, such a list would be quite useless as a means of communicating mathematics from author to reader and we shall not provide one here. What follows should be regarded instead as an informal specification of such a list: theorems will be stated in a mixture of the formal first-order language and informal English; we shall from the outset use definitions to introduce abbreviations, without which the statements of even the simplest theorems would be far too long to take in; and the ‘proofs’ will in fact be no more than attempts to indicate how a fully formalized proof could be constructed.
The axiom of choice
Authors: Potter Michael D
Published: November 2023

We return now to the consideration of the conception of set theory which our axiom system is intended to codify. We have already (in the note on ‘Indefinable collections’ in §1.2) remarked on the question whether we should allow the possibility of collections which are not, even in principle, definable. We left this question unresolved then, but in §5.5 we were forced, in proving that every set is finite or infinite, to assume the existence of a sequence which we could not explicitly define. The axiom which we used there-that of countable choice-is the prototype for the ones we shall be considering here. In particular, the assertion which we obtain by removing from it the assumption of countability-the so-called ‘axiom of choice’-is central in what follows.
The prime ideal property
Authors: Potter Michael D
Published: November 2023

In this chapter we shall show that (assuming the axiom of choice) every small distributive lattice L can be represented as (i.e. is isomorphic to) a sublattice of an appropriately chosen power set 𝒟 (spec(L)). This generalizes the results on the representation of spatial lattices which we obtained in §8.7. In the course of this endeavour we shall come across several lattice-theoretic assertions provably equivalent to the axiom of choice, and several (including the representation result itself) equivalent to a weaker principle called the ‘prime ideal property’.
Relations
Authors: McLarty Colin
Published: November 2023

Precisely, we define a quotient for an equivalence relation ⟨h,k ⟨ to be a coequalizer for h and k which has them as its kernel pair. Show that an arrow q:A → B is a coequalizer for some parallel pair into A iff it is a quotient for its own kernel pair.
Cartesian closed categories
Authors: McLarty Colin
Published: November 2023

This chapter assumes a base category with finite limits, and after defining exponentials we assume that every two objects have one. An intuitive discussion and examples follow in Section 6.5.
Product operators and others
Authors: McLarty Colin
Published: November 2023

This chapter deals with a formal issue. With products, for example, once we know that objects A and B have at least one product we pick one and call it A × B. Similarly, if B has at least one exponential by A we pick one and call it B’ As long as we deal with only a finite number of objects at a time this can be formalized in first order logic, with no axiom of choice, using no actual terms A × B or B. But if, for a given A, we want to once and for all pick one exponential BA for each B, that involves an axiom of choice. And even if only a finite number of choices are made there is a kind of awkwardness in them. There fore we adopt a device for avoiding them.
Mathematical foundations
Authors: McLarty Colin
Published: November 2023

The axioms for, say, a Cartesian closed category are a sufficient foundation by themselves for the theory of a single such category. No stronger background theory or metatheory is needed for the theorems of Chapter 6. Similarly, the axioms in Chapter 22 suffice as a formal foundation for a category of sets, and those in Chapter 23 for a category of smooth spaces. But the past few chapters have worked with many categories at once, and this requires some further foundation. In what follows, two styles of mathematical foundations for general category theory are described. The first, found in most of the literature, interprets category theory in some axiomatic set theory. The second, still being developed, axiomatizes a category of categories in categorical terms.
Soundness proof for topos logic
Authors: McLarty Colin
Published: November 2023

In this chapter the truth value false, negation, disjunction, and existential quantifiers are defined, and it is proved that topos logic is sound. A reader who feels no need for the details may omit this chapter.
Euclidean rings
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

The division algorithm for integers is enormously useful, and so it makes sense to investigate whether something similar and equally useful works, say, for Gaussian integers. When it works (as indeed it does for Gaussian integers), it can be used as the foundation for a successful extension, to the rings in question, of arithmetic ideas such as prime numbers and highest common factors. It is then possible to use these new arithmetics to solve problems in classical number theory and some other branches of mathematics. We shall give some of these applications in later chapters.
The four-squares theorem
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

We are now in a position to use the ring of Gaussian integers to prove one of the most famous theorems in number theory: every positive integer is the sum of four squares. The material of this chapter is a bit tricky and is not used later, so that the reader may omit it or read it superficially without worrying about the details. Nevertheless, we recommend that you look at Comments 4.7 and the Exercises. Let us set the scene by considering some typical problems of classical number theory. Because for the time being we are in number-theory mode, when we say ‘square’ we mean ‘square of non-negative integer’ and similarly for cubes, etc. There has been a great deal of interest in questions about whether every positive integer can be represented as the sum of such-and-such a number of special integers; or which integers can be represented in this way; or can every large enough integer be represented in this way. Here are some examples.
Unique factorization domains
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

At first it is perhaps rather difficult to appreciate why so much importance is placed on the result that every positive integer (except 1) is a unique product of prime numbers (see 3.14). It is known as the fundamental theorem of arithmetic, which suggests that it must be very significant, but in practice we take it for granted that, for instance, the only way of writing 10 as a product of prime numbers is as 2·5 (or 5 · 2, but that is really the same thing). We might hope that this sort of unique factorization property, when suitably defined, also holds in such systems as the Gaussian integers. If it does, we will have to prove it; we have no right to take it for granted. In fact we shall show that unique factorization does hold for Gaussian integers, but we shall also give examples (one of them apparently rather similar to the Gaussian integers) where unique factorization fails. It is these examples of failure which may bring home to us for the first time the need to be careful. Unique factorization should not be taken for granted, and when it does hold we should appreciate its value in underpinning some of the arithmetic arguments which we may want to use.
Algebraic numbers
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

Many important numbers, such as ../2 and i, are solutions of polynomial equations. Such numbers are said to be algebraic, and we shall study their basic theory in this chapter. Algebraic numbers also occur frequently in the following context: We start with a polynomial f(X) over Q and we wish to study the solutions of the corresponding equation f(X) = 0. These solutions are algebraic numbers. Definition 11.l. Let K be a field with a subfield F. Then t EK is said to be algebraic over F if f(t) = 0 for some non-zero polynomial f(X) with coefficients in F.
Ruler-and-compass constructions
Authors: Chatters A W , Hajarnavis C R
Published: November 2023

In the Greek tradition of geometry there has been much interest in determining which geometrical constructions can be performed using only two very simple devices. This question is asked in an idealized context where constructions can be performed with complete accuracy. The first permitted device is a ruler or straight edge; this makes it possible to draw the line joining distinct points A and B in the plane and to extend the line as far as necessary in either direction. The ruler has no markings on it and it cannot be used for measuring lengths. The second device is a pair of compasses; this makes it possible to draw the circle with centre at given point A passing through a second point B. It was known to the ancient Greek geometers that it is possible, using only these devices in combination, to perform such constructions as bisecting a given angle or dropping a perpendicular from a point to a line (details below).
Toposes - Basic Theory
Authors: Johnstone Peter T
Published: November 2023

We are now ready to present the definition of an (elementary) topos, which is the central notion of this book: it is a category having all the good ‘set-like’ properties discussed in the previous chapter, except (possibly) for Booleanness. It is a remarkable fact that all these properties are implied by the conjunction of just two of them:
Allegories
Authors: Johnstone Peter T
Published: November 2023

The object of this Chapter is to present an alternative approach to the notion of topos, different from (but not entirely independent of) the one which we followed in Chapters A1 and A2, via categories of relations. Whlle this appt'cach will not play a very important role in our subsequent exposition of topos theory, it is of some interest in its own right, and it certainly provides an appealing route to the construction of some of the leading examples of toposes, which can make their structure easier to understand.The advantages of this approach have been forcefully argued over some years by P. Feeyd, and a fuffer account of it wlll lee found in [381]. There are a number of other approaches to the study categories of relations - notably that developed by A. Carboni and ft. Wallers [322], whidi makes heavier use of 2-categorical ideas - but we shall not describe them here.
Many-particle systems
Authors: Lunn Mary
Published: November 2023

There are many binary systems of stars studied by astrophysicists. The binary system is particularly stable but true tertiary star systems are relatively rare. In fact tertiary systems often comprise a binary system together with a single star. There is no complete analytical solution to the equations satisfied by a tertiary system of star. For systems of more than two stars the equations have to be solved by numerical methods and stable solutions appear to be rare. The planetary system is of course stable but then the Sun is considerably more massive than any of the planets. We shall study the equations associated with many-particle systems very briefly.
An Introduction to Spectral Methods for Partial Differential Equations
Authors: Quarteroni A
Published: November 2023

This lecture is an elementary introduction to spectral methods for the numerical approximation of boundary value problems. The basic properties of Fourier and Chebyshev orthogonal expansions are revisited. The discrete Fourier and Chebyshev transforms are introduced, and it is shown how to use them in the framework of spectral collocation approximations. Several examples of applications are shown, including elliptic boundary value problems, advection-diffusion equations and Navier-Stokes equations. Both periodic and non-periodic boundary conditions are considered.
Univariat Splin S
Authors: Dierckx Paul
Published: November 2023

Spline functions (splines) are currently used in diverse domains of numerical analysis (interpolation, data smoothing: computer aided geometric design. the mmwrical solution of differential and integral equations, etc.). A landmark iu their increasing popularity has certainly been the publication of efficient algorithms for calculating with B-sp1ines (de Boor 1972; Cox 1972). The latter still form the basis of several high quality spline software packages.

In this first chapter we will collect the basic properties of splines and B-splines together for further reference. We will intuitively explain why the latter are so spf ‘cial and important. We will also give an impression of the computational complexity of the B-spline algorithms mentioned above. Exact proofs will be omitted. They can be found in the basic works of de Boor (1978) and Schumaker (1981).

A more recent spline topic is knot insertion. It plays a central part in interact in• CAD spline systems. It will also appear to be extremely useful in a numht>r of applications that we will consider. \Ve will therefore briefly commt>nt on knot insertion algorithms and indicate some connections with the other B-spline algorithms.
Curve Fitting: An Introduction
Authors: Dierckx Paul
Published: November 2023

In this introductory chapter we will state the curve fitting problem and briefly review a number of published spline solution methods. Reference will be made to particular software implementations of which we are aware. They are to be found in the appropriate chapters of the well known general software libraries (NAG (1988), IMSL (1987), PORT (Fox 1984), HARWELL (1987)) or in specific spline packages (fitpack (Cline 1974) and FITPACK ; Dicrekx 1987a. 1989). not to be confused, DASL (Anthony and Cox 1987b), SPLINE TOOLBOX for use in MATLAB (de Boor 1990)). A lot of these routines are also distributed freely through netlib (Dongarra and Grosse 1987); readers with an electronic mail facility are advised to consult the netlib libraries dierckx, fitpack, gcv, pppack and toms in particular (email address: netlib@oml.gov). Finally, we also refer to the paper by Grosse (1990) which contains an extended catalogue of algorithms for numerical approximation with references to the literature and pointers to available code
Controlling Reduce; operators and matrices
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

We begin with some technicalities of the interactive control of REDUCE, then consider how to manipulate polynomials and rational functions in more detail, how to perform substitutions and control simplification, and how to control the internal and external forms of expressions. Finally, we introduce two more structured data types: operators and matrices.
A look inside Reduce
Authors: MacCallum Malcolm A H , Wright Francis J
Published: November 2023

To understand what we find when we look inside an algebra system we must first have some idea what the data is that the system is representing. For emphasis, we repeat here that although the systems are performing calculus, they are not doing analysis; that is, they are emphatically not taking limits, testing convergence, and so on, except where those operations can be done by a strictly algebraic method: this is of course just what humans do when performing calculus. As mentioned in the first chapter, all of us will happily differentiate elementary functions without going back to first principles and taking limits, because we know the computational rules, like the rule for differentiating a product, which ensure that the results are correct, and we know that all the usual functions obey these rules. The rules are all in fact algebraic in nature.
Ranked Universe
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

Properly speaking, the book begins with this chapter. In Section 4.1 we define the notion of a ranked universe. We leave to the Appendix C and [285] the verification that the ranked groups are in fact the groups of finite Morley rank, as the original definition of Morley rank plays no particular role here. The key element of the definition is Axiom A, a noetherianity condition. We also state as Conjecture 4.7 the main conjecture, discussed in the Preface, which has motivated most of the work discussed here. Section 4.2 contains the first consequences of the definition, and for many of them it may be said that if they did not follow from the definition given, we would be forced to consider them as additional axioms. (Section B.5 contains a discussion of some desirable properties that may not follow from the definitions.) Finally Section 4.3 introduces the more technical notion of uniform definability, corresponding to the notion of an algebraic family of algebraic varieties in algebraic geometry. The exercises for Section 4.2 will enable the reader not accustomed to the model theoretic point of view to develop a better appreciation of the content of the formalism presented, and at the same time to check his or her understanding of the notions presented in Chapters 2 and 3. The first substantial results concerning groups of finite Morley rank (ranked groups) are found in the next chapter.
Bad Groups
Authors: Borovik Alexandre , Nesin Ali
Published: November 2023

A remarkable result about algebraic groups asserts that the Borel subgroups (i.e. the maximal connected solvable closed subgroups) of an algebraic group Gover an algebraically closed field are conjugate to each other (see e.g. [189, §21.3]). Moreover, if G0 is nonnilpotent, then its Borel subgroups are also nonnilpotent [189, §21.4]. The situation in groups of finite Morley rank is not so well-understood. We can introduce the notion of a Borel subgroup of a group G of finite Morley rank in almost the same words: B :S G is a Borel subgroup if B is maximal among the connected definable solvable subgroups of G. However the proof of the conjugacy of Borel subgroups in algebraic groups cannot be generalized to the context of groups of finite Morley rank, because the former is based on the notion of complete variety from algebraic geometry, which does not have a model theoretic analogue.
Möbius Transformations and Inversion
Authors: Needham Tristan
Published: November 2023

Thus, although more than 150 years have passed since August Ferdinand Mobius first studied the transformations that now bear his name, it is fair to say that the rich vein of knowledge which he thereby exposed is still far from being exhausted. For this reason, we shall investigate Mobius transformations in considerably greater depth than is customary.Clearly it would be neither appropriate nor feasible for us to explore this connection in detail, but let us at least briefly indicate how Möbius transformations are related to Einstein’s Theory of Relativity.
Further Geometry of Differentiation
Authors: Needham Tristan
Published: November 2023

In the previous chapter we began to investigate the remarkable nature of analytic functions in C by studying mappings in the less structured realm of ℝ. In particular the Jacobian provided us with a painless way of deriving the Cauchy-Riemann characterization of analytic functions, and also of computing their amplitwists. However, this approach was rather indirect. In this chapter we will instead study differentiation directly in the complex plane, primarily through the use of infinitesimal geometry. Our first application of this approach will be the rederivation of the Cauchy-Riemann (henceforth “CR”) equations, and the discovery of new forms that they can take on.
Winding Numbers and Topology
Authors: Needham Tristan
Published: November 2023

In this chapter we shall investigate a simple but immensely powerful concept—the number of times a loop winds around a point. In Chapter 2 we saw that this concept was needed to understand multifunctions, and in the next chapter we will see that it plays an equally crucial role in understanding complex integration. However, only the first two sections [up to (2)] of the present chapter are actually a prerequisite for that work; the rest may be read at any time. If you are in a rush to learn about integration, you may wish to skip the rest of the chapter and return to it later.
How to silence a universal machine
Authors: Smullyan Raymond M
Published: November 2023

As further applications of diagonalization, we now turn to a fascinating variety of problems that generalize some results in recursion theory known as halting problems. That they really do generalize theorems in recursion theory will be seen in a later chapter. The results of this chapter are also related to incompleteness theorems, as will be seen in the next chapter.
Epistemic Aspects of Databases
Authors: Winslett Marianne
Published: November 2023

This chapter explores the potential use of logic, and of the belief revision techniques of the preceding chapter, in databases and knowledge bases. In Section 1, we show how to formulate relational databases in first order logic, and discuss the use of logic in studying database queries, updates, incomplete information, and integrity constraints. Section 2 concentrates on the problem of updates to knowledge bases, a generalization of databases in which integrity constraints and incomplete information may combine to make updating very difficult. Section 3 shows how logic can be applied to the problems of diagnosing faults in circuits and of reasoning about the effects of actions.
Temporal Logic
Authors: Benthem J van
Published: November 2023

There are various historical sources for the discipline of Temporal Logic. The insight that temporal discourse and temporal argument show significant logical structures arises naturally in the empirical study of human reasoning, but it has recently become a necessity in the design of mechanical reasoning systems as well. A few specific examples may serve to start off our survey.
Towards a Computational Treatment of Time
Authors: Gabbay D M , Reynolds M
Published: November 2023

This chapter is about topics relevant to the algorithmic 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. This is not difficult to do. Time is a very exciting subject. It is one of the few subjects on which everyone is an expert. We move through time continuously and in order to survive and manage ourselves sensibly we constantly have to make temporal decisions. Philosophy, since the days of Aristotle, has been trying to analyse the way we make these decisions. With the rise of computer science, where ideally one wants the machine to do the job for or of the human, there is a new urgency in the precise logical analysis of human temporal activity.
Non-monotonic Temporal Reasoning
Authors: Sandewall Erik , Shoham Yoav
Published: November 2023

Temporal reasoning is not restricted to reasoning about time itself; it also includes reasoning about phenomena that take place in time, i.e. reasoning about actions and change. Such reasoning is required for a number of purposes in Al programs, in particular for prediction, planning and plan modification, plan recognition, and diagnosis. These types of reasoning tasks depend crucially on the use of action laws, i.e. axioms which specify what effects an action will have or may have in a particular situation. Besides the action laws, each reasoning task typically involves (1) designation of certain actions which have been (will be, may be) performed, as well as their order of execution; (2) statements about the state of the world before the actions; (3) statements about the state of the world after the actions. In a planning task (2) and (3) are given and (1) is sought; in a prediction task (1) and (2) are given and (3) is sought, and so on.
Congruence axioms
Authors: Roe John
Published: November 2023

Affine geometry by itself does not capture the full richness of our intuitive picture of geometry. Its chief deficiency is that we are only able to compare lengths along the same line by means of the ruler axiom; and though we can extend this to compare lengths along two parallel lines (by drawing appropriate parallelograms), it makes no sense in affine geometry to compare lengths AB and AC if AB and AC are not parallel.
The Mathematical Study and Approximation of Semi-conductor Models
Authors: Jerome Joseph W
Published: November 2023

One may view classical or semi-classical semi-conductor modelling as a pyramid structure, in which the Boltzmann Transport Equation (BTE) forms the summit and the drift-diffusion system the base; intermediate are systems derived by taking moments of the Boltzmann equation. The hydrodynamic model and its momentum relaxation time limit, called the energy model, are examples. These will be discussed more fully in the final technical section of this chapter.
The rational line
Authors: Huxley M N
Published: November 2023

The rational numbers form a countable set, dense in the real numbers. They are set in the real line like stars in the sky ‘to illuminate the mystery of the continuum’ (a remark attributed to Borel). In any region, if magnified sufficiently, faint stars appear. To explain this bright and faint analogy, we need the concept of height.
The integer points close to a curve
Authors: Huxley M N
Published: November 2023

When δ is small, then the estimates for rounding error sums are larger than δL, and we can hope only to find an upper bound for R. When δ = 0 we are back to counting points on the curve. Suppose that a set of points contains far more integer points than one would expect.
Mean value results
Authors: Huxley M N
Published: November 2023

The idea in this chapter is to investigate a sum S by introducing a real parameter x, and then evaluating Apart from its own interest, the integral has three uses.
The simple exponential sum
Authors: Huxley M N
Published: November 2023

Estimating the sum S is analogous to lattice point questions for enlargements of a fixed curve. The relevant curve is not y = F(x), but y = F'(x).
Exponential sums with modular form coefficients
Authors: Huxley M N
Published: November 2023

Modular forms, and their generalization to automorphic functions, lie at the centre of mathematics, in the sense that they appear in combinatorics, group theory, number theory, algebraic geometry, geometric topology, functional analysis, and complex analysis. Swinnerton-Dyer once declared that the unsolved problems in mathematics remain unsolved because we have not yet connected them to modular forms.
The Hardy-Littlewood method
Authors: Huxley M N
Published: November 2023

The Hardy-Littlewood method is simplest when there is only one condition defining the counting function on the left, and only the integration over a on the right. It has a divide-and-conquer structure.
The First Spacing Problem for the double sum
Authors: Huxley M N
Published: November 2023

The inequalities (13.1.2) and (13.1.4) are different from Chapter 11, and the range (13.1.5) is more restricted. We shall not use condition (13.1.4). We follow Watt’s account (1990).
Exponential sum theorems
Authors: Huxley M N
Published: November 2023

We resume the treatment of the simple exponential sum in one variable. from Chapter 7, obtaining the results of Huxley (1993a), extended to short intervals and congruence families. We use the notation of Chapter 7, and assume all the restrictions made there on the parameters N (length of short sums) and R (expected size of the denominator). We have estimated the numbers A and B in Lemma 7.4.3.
Related work
Authors: Huxley M N
Published: November 2023

Bombieri and Pila (1989) have improved Swinnerton-Dyer’s estimate for the number of points on a convex curve when the curve is very smooth. The argument again takes a ‘divide and conquer’ form.
Symmetric Domains Of Tube Type
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

The tube domains considered in the preceding chapter are multidimensional analogues of the upper half-plane in C. In this chapter, for the case of tubes over symmetric cones, we introduce the analogue of the fractional linear transformation which maps the upper half-plane onto the unit disc; in the special case of the matrix domains this map coincides with the classical Cayley transformation. We give a description of the image domain, its kernel function, its Shilov boundary, its Bergman metric, and its group of holomorphic automorphisms.
Conical And Spherical Polynomials
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

As before, in this chapter V will be a simple Euclidean Jordan algebra, Vℂ its complexification; we will asume that the inner product on Vℂ is given by (z/w) = tr(zw). We will denote by 𝒫 the space of all polynomials on Vℂ. When convenient, we will also think of 𝒫 (without changing notation) as the space of complex-valued polynomials on the real vector space V. A natural representation π of the group GL(Vℂ) is defined on 𝒫 by
Taylor And Laurent Series
Authors: Faraut Jacques , Korányi Adam
Published: November 2023

In this chapter we use the results developed in Chapter XI to study series expansions of functions naturally associated with the geometry of a symmetric cone. In the first section we consider the important special case of series representing K-invariant analytic functions in a neighbourhood of the origin. In subsequent sections we define Laurent expansions for functions analytic in U-invariant domains in Vℂ and we extend the theory of expansions to include functions that are not necessarily K-invariant. We also describe the hull of holomorphy of U-invariant domains.