bluuebunny/crossref_metadata_2025 huggingface dataset:
Bishop's condition ( 13) and invariant subspaces
Authors: Eschmeier Jörg
Published: November 2023

The main idea of the homological approach to the analytic functional calculus problem is to identify a given commuting system a ε L(X)n of continuous linear Banach space operators with the σ(ℂn)-module structure σ(ℂn) ×X→X→X, (f,x)↦f(a)x determined by a, and then to extend this module structure to the algebra Ma (a)) of all germs of analytic functions defined near the spectrum a (a) of a, applying methods from topological homology theory.
Applications to function theory
Authors: Eschmeier Jörg , Putinar Mihai
Published: November 2023

The abstract notions of quasi-coherence, sub-scalarity, analytic transversality and so on have been related to certain concrete problems which we met in the preceding chapters. These examples offered both a motivation and a validation of the whole theory. The present chapter is intended to bring forward some further examples and applications related mainly to function theory of real or complex variables. This terrain abounds in localization methods and transversality phenomena. Their homological interpretation, based on the preceding chapters, provides a new insight into some old and new results, usually obtained by laborious methods.
Irregularities in the Distribution of Primes
Authors: Friedlander John B
Published: November 2023

In this paper, we survey a body of results, stemming from an idea of Maier [M], which has demonstrated that the distribution of primes in certain basic integer sequences is not as uniform as it was once believed to be. We also make some speculations, both as to further problems that might be amenable to these methods, and as to the nature of some (still apparently) reasonable conjectures to modify those which have been found no longer tenable.
Cubic Inequalities of Additive Type
Authors: Brüdern J , Cook R J
Published: November 2023

The first result on cubic inequalities is due to Davenport and Roth [7]. They proved that if λ1,...,λ8 are real numbers with at least one ratio λi/λj irrational then for any ε > 0 the inequality has infinitely many solutions in integers x1,..., x8. No progress has been made on reducing the number of variables, however the result can be made more precise.
Paulo Ribenboim, at the time of his retirement
Authors: Granville Andrew
Published: November 2023

At this, the third meeting of the Canadian Number Theory Association, we have had the opportunity to enjoy a lot of good mathematics, while celebrating the career of one of Canada’s most distinguished academics. Paulo Ribenboim, author of more than a hundred journal articles and of thirteen books, a Fellow of the Royal Society of Canada, and colleague and advisor to so many mathematicians, is retiring after a career spanning over forty years and three continents. In this article we present a biography of Paulo, together with a sampling of a few of his many interesting results.
The Kummer-Wieferich-Skula Approach to the First Case of Fermat’s Last Theorem
Authors: Granville Andrew
Published: November 2023

In 1909 Wieferich surprisingly deduced, from Kummer’s criteria, that p2 must divide 2P - 2. Soon others deduced that p2 must also divide 3P - 3, 5P -5 and so on; and Frobenius outlined a method to continue proving such criteria.
Images and patterns
Authors: Grenander Ulf
Published: November 2023

In order to analyse and understand a regular phenomenon in nature or in the man-made world we try to describe it in terms of rules, laws, regularities. If we formalize the concept of regularity as was done in the preceding chapter we arrive at the regular structures generally denoted by 𝒞(𝒮), the configuration spaces. A configuration space is a mathematical abstraction that expresses how simple objects, the generators, are combined to form complex systems, the configurations. The combinatory rules leading to 𝒞(𝒮) must now be supplemented by rules of interpretation that express the relation between the configuration, as an abstraction, and what can be observed concretely, seen, heard, measured, and so on.
Abstract biological patterns
Authors: Grenander Ulf
Published: November 2023

Biological growth patterns exhibit so much variability, such dramatic developments, and such a formidable array of possibilities that full understanding of them will probably remain an elusive goal for a long time to come. Attempts to achieve some degree of understanding reach far back; to antiquity as far as descriptive and taxonomic studies are concerned, and to the turn of the century in the case of laboratory work with grafting and surgical experiments (see Note 4.A).
Patterns of collective behaviour
Authors: Grenander Ulf
Published: November 2023

In the dynamics in Chapter 4 the connector graph changed from one subgraph of a lattice to another: the graphs were embedded in a fixed graph, a lattice, which simplifies both the analysis and the algorithms considerably. It is clear, however, that this assumption — all σ’s embedded in a fixed graph — is highly restrictive. We now turn to pattern dynamics where we encounter more essential changes in connector structure, which will lead to some challenging difficulties.
Non-Bayesian pattern inference
Authors: Grenander Ulf
Published: November 2023

In the absence of prior measures on the regular structures we have to use methods of classical statistical inference, such as Neyman-Pearson testing, maximum likelihood and least squares estimation. If the configuration space/image algebra has no detailed structure, say it is just a finite-dimensional Euclidean space with no added relations, then the application of classical statistics need not involve any new ideas. Such cases will not be discussed here where the emphasis is on structured pattern analysis, where the models are intended to incorporate more of the characteristic features of the images under study. We do not get the increased realism of the model free ߝ the price we pay is sometimes considerable in terms of analytical/computational complexity. One such difficulty is caused by the high dimension of the regular structures. This can invalidate standard procedures; maximum likelihood estimation may not even be consistent.
Pattern recognition
Authors: Grenander Ulf
Published: November 2023

The uniform approach to pattern inference developed in Chapters 13-17 enables us to deal with the different tasks - restoration, extrapolation, recognition, etc. - by the same methodology. We have, for example, developed inference algorithms in the same spirit for image restoration, extrapolation, and detection of abnormalities, and shall now show how this can be done for recognition. We have already treated a number of recognition tasks, especially in Chapter 16, but will now study some where the restoration aspect is only secondary.
Creating acceptor functions and connectors
Authors: Grenander Ulf
Published: November 2023

Assume that the connection type Σ contains only graphs without cycles. Then we can sometimes estimate the acceptor functions, or other parameters determining the regularity controlled probability measure, by direct methods; we do not encounter the difficulty just mentioned. To illustrate the technique let us consider a context free language (see Sections 1.3.5 and 8.1.2) assuming for simplicity that it be unambiguous in the sense that each sentence has an essentially unique derivation tree (Note 20.1).
Symmetry
Authors: Woodhouse N M J
Published: November 2023

Symmetry principles play an important part in theoretical physics, both as a means of understanding conservation laws and as computational devices providing quick and elegant solutions to problems that would otherwise be tedious to solve. In quantum theory, their role is especially significant since they not only clarify basic concepts and simplify the formal development of the subject, but they also provide detailed information about physical systems for which no explicit dynamical models exist. However, the methods used to investigate symmetry in quantum systems are often based on abstract group-theoretical techniques, which bypass the canonical formalism. They include no general rules for going over to the classical limit and for directly relating the symmetries of the quantum system to those of a corresponding classical system.
Complex Polarizations
Authors: Woodhouse N M J
Published: November 2023

Real polarizations play an important part in linking classical and quantum mechanics. We shall see that picking a polarization in a classical phase space amounts to choosing a representation in the underlying quantum theory-’representation’ in the sense of a position or momentum space representation. We shall also come across complex representations, which in many ways are easier to handle. An obvious example is the Fock space treatment of the harmonic oscillator, in which the quantum Hilbert space is realized as a space of holomorphic functions Complex representations arise from complex polarizations, which are natural generalizations of real polarizations: one simply relaxes the condition that the Hamiltonian vector fields spanning P should be real.
Quantization
Authors: Woodhouse N M J
Published: November 2023

The problem with prequantization is that the Hilbert space it constructs is too large to represent the phase space of a physically reasonable quantum system. In the first example in §8.4 (a cotangent bundle), the Hilbert space contained wave functions that depended on momentum as well as position; and in the third (a Coadjoint orbit), the group representation was reducible. We have already touched on a possible solution: we should admit only sections of B that are parallel along a polarization. We shall now look in detail at how this works. It should be stressed, however, that the physical justification is not based on general mathematical results (such as the Borel-Weil theorem), but on the way in which the construction works in particular examples. It generalizes and unifies a number of quantization techniques that, in the past, have not appeared to have any obvious connection with each other and that have sometimes seemed overspecialized with applications only to particular physical systems. Also it should be noted that not all syrnplectic manifolds admit polarizations (Gotay 1987).
The Metaplectic Correction
Authors: Woodhouse N M J
Published: November 2023

The half-form bundles are also defined for complex polarizations. By including them in the quantization construction, we arrive at a more useful pairing, which can be applied to a much wider class of nonnegative polarizations. It seems to give the correct physical relationship between the wave functions in the Hilbert spaces of different polarizations. In the case of two Fock spaces, the new pairing is unitary. When P is complex, δp may be nontrivial, and its inclusion in the construction can modify quantization conditions and alter the form of the quantum observables. In the case of the harmonic oscillator, the modification is precisely that needed to get the correct zero-point energy (see §9.2).
Hall bases
Authors: Reutenauer Christophe
Published: November 2023

Hall sets of trees are introduced in Section 4.1. A rewriting system on standard sequences of Hall trees is defined and shown to be confluent, invertible, and always terminating. As a consequence, each word has a unique decreasing factorization into foliages of Hall trees. By interpreting the nodes of a Hall tree as the Lie bracket, one obtains a basis of the free Lie algebra, and the corresponding Poincare-Birkhoff-Witt basis, in Section 4.2. In the last section it is shown that this basis construction gives exactly the same bases as those obtained by elimination in Chapter 0. In the appendix it is shown amongst other things that Hall bases are in some sense optimal.
The action of the symmetric group
Authors: Reutenauer Christophe
Published: November 2023

In Section 8.1 we present the duality between the representations of the symmetric group and the linear group. From this, we quickly deduce the character of the free Lie algebra, already knowing the generating functions of primitive necklaces. This character is induced by a faithful one-dimensional representation of the subgroup generated by a circular permutation (Section 8.2). In Section 8.3 we give the combinatorial interpretation of the multiplicities of irreducible representations in the Lie representation. It is also shown that almost all irreducible representations actually appear. Section 8.4 introduces remarkable Lie polynomials, the Lie idempotents. In the last section, we consider the representations arising from the canonical decomposition of the free associative algebra.
Continuous Wavelet Transform
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

The objective of this chapter is to generalize the important subject of the continuous wavelet transform to the spherical case. The spherical continuous wavelet transform arises naturally as a result of an integral reproducing formula that is closely related to the theory of singular integral operators. In the spherical continuous wavelet transform the rotation and dilation parameters vary continuously, i.e. such a reconstruction requires global information at allscales ρand positions η.
Vectorial Approximation Methods
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

This chapter combines scalar constructive approximation theory with the vectorial concept of spherical harmonics. It turns out that the presented notation and basic settings lead in canonical way to vectorial generalizations of Sobolev spaces, pseudodifferential operators, radial basis functions, splines, singular integrals, wavelets, etc. It should be noted, however, that we have made no attempt to be comprehensive in the coverage of constructive vectorial approximation. Only the cornerstones are mentioned, to enable readers to extract more specialized information from the scalar part of this book for themselves.
Postscript: A Broader, not Only Professional View
Authors: Kreisel Georg
Published: November 2023

The material below may be of use to some of those readers who have gone through the paper above, and have its mathematical ideas at their fingertips; in whatever sense this phrase is appropriate, to temperament and education of a particular reader. Beyond this, as a corollary to a simple, but important home truth repeated since Pindar, the readers should not — even feel to be — so wise that the facts are enough for them. The material is concerned with an interpretation, in particular, by means of relations to broad proof-theoretic experience. Since it is not only possible, but common, to assume that scholarly traditions are suitable for such an enterprise, a general disclaimer is necessary here; to be bolstered by specifics when occasions arise.
Cox processes
Authors: Kingman J F C
Published: November 2023

In Section 5.5 we met a random set which, while not a Poisson process, could be made into a Poisson process by conditioning. Such processes were introduced by Cox (1955) under the name of doubly stochastic Poisson processes, but are now usually given the name of their discoverer.
The Poisson-Dirichlet distribution
Authors: Kingman J F C
Published: November 2023

The parameters α j may take any strictly positive values, and the character of the distribution changes markedly as these vary. If α j = 1 for all j we have the uniform distribution on Δ n If the α j = are large, (9.3) concentrates probability well away from the boundaries of Δ n, corresponding to distributions p which are fairly evenly spread. On the other hand, small values of the α j = concentrate near the boundary of Δ n, corresponding to highly disparate p with a few large p   j and the others small.
Balance of Forces; Power
Authors: Gurtin Morton E
Published: November 2023

Consider an evolving interface 𝓈 with r a corresponding parametrization. The micromechanics of the interface is described by three interfacial fields:

C(s, t) capillary force,

B(s, t) interactive force,

m(s, t) interactive moment.
Stationary Interfaces and Steadily Evolving Interfaces
Authors: Gurtin Morton E
Published: November 2023

We assume throughout this chapter that the interfacial energy is strictly stable, so that

This chapter is restricted to evolving interfaces that are admissible; to avoid repetition, we will omit the term ‘admissible’ in most of the ensuing discussion.

Evolving interfaces are, by definition, smooth, a limitation that involves no loss in generality, since the interfacial energy is strictly stable (cf. (7.30)).
Unstable Interfacial Energies and Interfaces with Corners
Authors: Gurtin Morton E
Published: November 2023

Suppose now that

the interfacial energy is regular, hut not stable

(cf. the last paragraph of Section 7.3). Then an admissible evolving interface must, by virtue of its smoothness, involve normal angles at which the evolution equations are backward parabolic. Two ways of overcoming this difficulty are to:

regularize the evolution equations;

allow corners that correspond to jumps in 0 across the unstable portions of .f(θ).
Non-Smooth Interfacial Energies; Crystalline Energies
Authors: Gurtin Morton E
Published: November 2023

Material scientists often consider interfacial energies that are continuous but have derivatives that suffer jump discontinuities. We now discuss a particular class of energies with this property. These energies, termed crystalline, have convexified Frank diagrams that are polygonal. As we shall see, crystalline energies are compatible with fully faceted interfaces.

We assume that the energy f(θ) is crystalline; precisely, we assume that f(θ) is continuous, piecewise-smooth, strictly positive, and consistent with the following two conditions.

Material scientists often consider interfacial energies that are continuous but have derivatives that suffer jump discontinuities. We now discuss a particular class of energies with this property. These energies, termed crystalline, have convexified Frank diagrams that are polygonal. As we shall see, crystalline energies are compatible with fully faceted interfaces.

We assume that the energy f(θ) is crystalline; precisely, we assume that f(θ) is continuous, piecewise-smooth, strictly positive, and consistent with the following two conditions.
Instabilities Induced by Supercooling the Liquid Phase
Authors: Gurtin Morton E
Published: November 2023

Problems involving supercooled liquids and superheated solids are inherently unstable. In fact, for a crystal in a supercooled melt, it is precisely the unstable nature of the melt that induces growth of the crystal.
Solution adaptivity with structured grids
Authors: Catherall D
Published: November 2023

This paper considers the problem of solution-adaptive meshes from the point of view of mesh movement, rather than the currently more popular methods involving the embedding of regions of finer mesh. These latter methods are, to some extent, easier to apply, since they affect a mesh only locally, whereas as will be seen, with mesh movement methods the whole mesh is adapted, introducing global problems of mesh quality. In the concluding remarks at the end of the paper a short discussion is included on the relative merits of the two approaches.
Convergence acceleration for first order systems
Authors: Gustafsson Bertil , Per LÖtstedt
Published: November 2023

Any finite difference, volume or element approximation of a steady state flow problem gives rise to a large system of nonlinear algebraic equations. A direct solver for the linear systems arising from Newton’s (or similar methods) requires too much CPU-time and too much memory even on the fastest supercomputer for real problems in three space dimensions. It has been tried for 2-D problems in [1], but even for such problems it can hardly be considered as a general purpose method.
Iterative defect correction and multigrid accelerated explicit time stepping for the steady Euler equations
Authors: Koren Barry
Published: November 2023

The computational grid is obtained by a hybrid finite element - finite volume partition. A (possibly unstructured) finite-element triangularization is used as the basic partition. A cell-centered finite-volume partition is derived from the finite-element partition by connecting the centers of the triangle sides in the manner illustrated in Figure 1. The finite-volume grid gives us the easy possibility of grouping together the nodes associated with contiguous finite volumes.
Adaptive concepts for the resolution of compressible flow by upwind finite volume methods
Authors: Sonar Th. , Hannemann V , Friedrich O
Published: November 2023

The details of the scheme will be described in a forthcoming paper elsewhere. The construction outlined leads to a robust and accurate scheme capable of computing strong shocks as well as contact discontinuities. Figure 1 shows the density distribution in the Mach 3 forward-facing step example of Woodward and Colella [5]. The grid used consists of 14297 points and the result shown was obtained at time t=4 (using the same non-dimensionalisation as Woddward and Colella). In [5] a cartesian grid with more than 19000 points was used to compare several high-resolution finite-difference schemes. As can be seen by comparison with [5] the result of our scheme compares quite well with the result of high-resolution finite-difference methods.
Local uniform mesh refinement and brine transport in porous media
Authors: Trompert Ron
Published: November 2023

An adaptive-grid finite-difference method is applied to a model for unsteady, isothermal, coupled flow and transport of brine in porous media. In the vicinity of rock salt formations, the salt concentration may become large giving rise to disparate scales in the salt concentration profiles. A typical situation one encounters is that of a sharp fresh-salt water interface that moves in time. For such problems, a single uniform space grid held fixed throughout the entire time evolution can be computationally very inefficient, since, to afford an accurate approximation, such a grid would easily have to contain an excessive number of nodes, particularly so for two and three space dimensions.
Orthogonal block structured surface grids
Authors: Arina R , Tarditi S
Published: November 2023

One of the main issues in grid generation is the improvement of the grid quality, both for structured and unstructured meshes. In the present work we focus our attention on the generation of structured grids. It is important to develop structured grid generation systems able to give “good” grids in topologically complex regions, because in this way it is possible to reduce the effort of block decomposition, with considerable improvement of the flow solver efficiency. By “good” meshes, we mean grids sufficiently smooth, and orthogonal along part of the boundary (for example along solid walls). Moreover, along the interior boundaries connecting different blocks, it is advisable to have the possibility of controlling both the grid point positions and the slope of the coordinate lines. The aim of this work is to present a mesh generation method for structured grids, able to solve the main issues pointed out above, specifically: a) smoothness, b) orthogonality control, c) specification of the grid point position and coordinate line slope along each boundary at the same time.
Grid quality: an interplay of grid and solver
Authors: Veldman A E P , Botta E F F
Published: November 2023

Currently, much attention is being paid to the development of adaptive grid generation methods, which generate grids that are highly nonuniform. Firstly, this is motivated by the expectation that a high density of grid points is only necessary in regions of large solution activity (steep gradients, large curvature, etc.), whereas in the smoother regions of the solution larger grid cells can be used. Secondly, and more important, it is believed that a reduction in the number of grid points will also lead to a reduction of computational effort.
A high resolution discontinuous Galerkin method for hyperbolic problems on unstructured grids
Authors: F. Bassi , S. Rebay , M. Savini
Published: November 2023

In recent years successful results have been obtained in the numerical solution of Euler equations by means of total variation stable finite difference (volume) schemes. These methods, which in the earliest versions were restricted to first order accuracy and to one dimensional problems, have been successively extended to higher-order accuracy and, despite the lack of a rigorous mathamatical theory, to multidimensional problems. These extensions have been obtained by using wider stencils to increase the order of the numerical approximation and by means of one dimensional algorithms applied in prescribed directions to handle multidimensional problems. In practice, due to the presence of points aligned with logical directions, this approach is straightforward only when using structured grids. However, it is well known that structured grids suffer from heavy limitations when used for problems characterized by complicated geometry or when the grid has to be adapted to local features of the solution. For this reason in recent times a lot of research efforts have been devoted to the development of computational techniques suited to unstructured grids. Unfortunately, in the unstructured case the extension of higher-order one dimensional numerical schemes is by no means an easy task. Some attempts in this direction can be found for example in the work of Barth
On the steady-state accuracy of an implicit Navier-Stokes solver of Lax-Wendroff type
Authors: Lacombe G , Lerat A
Published: November 2023

When solving the Euler and Navier-Stokes equations, a key point is to avoid numerical instability and spurious oscillations without adding too much numerical dissipation. To overcome this difficulty for steady aerodynamic problems, an implicit unsteady method of Lax-Wendroff type has been developed at ENSAM and ONERA (Lerat 1983; Lerat and Sides 1988; Hollanders et al. 1983; Khalfallah et al. 1991; Lacombe and Lerat 1992). The original feature of this method is to produce efficient 2D and 3D flow calculations without artificial viscosity, upwinding or any correction (except in the hypersonic regime where some corrections may be required).
A solution method for the shallow water Quasi-equilibrium equations on the sphere
Authors: Mawson Mark H
Published: November 2023

A fundamental relationship in the atmosphere is the balance between the zeroth order approximation to the wind field and the pressure gradient force. In qualitative terms the wind field blows at right-angles to the pressure gradient. Current models of the atmosphere can be derived from the Navier-Stokes equations by making certain simplifying assumptions appropriate to the scales of motion current computer power allows us to resolve, typically 100km in the horizontal and 1km in the vertical, but do not include this fundamental relationship explicitly. Here we seek to develop a numerical method to solve a set of equations in which this property is enforced, namely the Quasi-equilibrium equations. The equations are obtained by adding this extra constraint into the usual set used in atmospheric modelling. A discussion of the possible benefits of using these equations can be found in (Cullen sl et al. 1987). A procedure for solving the three¬dimensional form of these equations on the sphere was given in (Mawson sl et al. 1990) but the method proved unstable in polar regions. Here we develop a solution procedure for the shallow-water form of the equations which remains stable.
Contamination of laminar boundary layers by artificial dissipation in Navier-Stokes solutions
Authors: Allmaras Steven R
Published: November 2023

Recent grid refinement studies conducted at Boeing for 2-D transonic airfoils revealed poor grid convergence characteristics for laminar boundary layers predicted by Navier-Stokes solvers using central differencing with scalar artificial dissipation (Jou et al. 1992). The studies showed that as the grid is refined, the skin friction in turbulent regions of the airfoil boundary layer becomes grid-converged before it does in laminar regions. These results were disturbing for two reasons. First, the laminar boundary layer regions had adequate resolution based on grid requirements for accurate solutions in boundary layer equation solvers. Second, the same Navier- Stokes solvers are known to perform well in low Reynolds number flows (Re ∼ 0(1000)).
Multigrid convergence acceleration of an upwind Euler algorithm on multiply-embedded meshes
Authors: Onslow S H , Albone C M
Published: November 2023

A multigrid convergence-acceleration scheme is derived and implemented within an upwind Euler method on three-dimensional, multiply-embedded Cartesian meshes. The multigrid scheme is a generalisation to embedded meshes of an earlier two-dimensional scheme due to Albone (1985) and is enhanced through the use of upwind-biased interpolation. The Euler algorithm is based upon the split-coefficient matrix method of Chakravarthy et al (1980) in its first-order accurate form. It is made spatially second-order accurate through the use of a defect-correction scheme which is a generalisation of an earlier scheme due to Albone (1988). The multiply-embedded Cartesian meshes are generated within the Feature Associated Mesh Embedding (FAME) method of Albone and Joyce (1990).
A new approach to finite element error control for time-dependent problems
Authors: Jimack P K
Published: November 2023

The vast majority of time-dependent partial differential equations (p.d.e.’s) cannot be accurately solved using numerical methods which do not permit some form of adaptivity in either in space or time. For most problems in CFD it is clear that both spatial and temporal adaptivity will be essential if we are to be able perform useful numerical simulations for a reasonable computational effort.
Recursion Theory
Authors: Phillips I C C
Published: November 2023

Computability theory might be a better name for recursion theory, since it is the mathematical study of what is computable. However the name is traditional. It arose because recursion theory started as the study of the recursive functions, a mathematical characterization of the functions which are computable (by some program). ‘Recursive’ because, just as in most programming languages, recursion is an important and very powerful mechanism for constructing these functions.
The Role(s) of Logic in Artificial Intelligence
Authors: Israel David J
Published: November 2023

From the very beginnings of research in artificial intelligence (AI), logic has played a central role. Rather, it has played a number of important roles. More recently, AI has begun to present exciting new challenges and open new lines of inquiry for logic. In this essay, we shall briefly sketch some of the roles logic plays in Al; we shall end with some mention of these recent challenges and lines of inquiry. This essay is not meant as a survey. The Handbook as a whole serves that purpose admirably; as, for each area covered, does each of its parts. Nor is it an editorial introduction to this volume or to the Handbook as a whole; there will, of course, be pointers into the Handbook where relevant. There are many aspects of the relation between work in logic and research in Al that we shall not discuss here. Most of these are dealt with extensively in the Handbook, many more than once.
Modal and Temporal Logics
Authors: Stirling Colin
Published: November 2023

In achieving their goals, programs manipulate data and thereby may change th state of the computer. Regarded as a dynamic system, the computer responds to program instructions and in so doing may change state. Activity and change are formalized by the operational semantics of programs and systems. Modal and temporal logics can directly express activity with their paradigm operators such as [a) (box a) when a is an action, and F. Where Φ is a proposition, [a)Φ expresses after every a action Φ holds while FΦ expresses eventually Φ holds. The modal operator [a] highlights the action that may provoke change of state whereas the temporal operator F focuses on the resulting changes. In both cases these operators may express properties of events that happen during a run of a system, events that may be crucial to understanding its overall behaviour. It is no accident that these logics can be used to express such notions. For their models comprehend structures that are encountered in operational semantics of programs and systems. These structures, (labelled) transition systems, are pivotal to this chapter.
Labelled deduction presentation of temporal logics
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

This chapter develops proof theory for temporal logic within the framework of labelled deductive systems [Gabbay, 1996).

To motivate our approach consider a temporal formula a = FA A. PB A. C.

This formula says that we want A to hold in the future (of now), B to hold in the past and C to hold now. It represents the following temporal configuration:

However, by far the simplest approach is to allow names of points in the language and write s : A to mean that we want s F A to hold. Then we can write a theory 6. as

the above ordering relations hold and the respective formulae are satisfied in the appropriate points.

The above language has turned temporal logic into a labelled deductive system (LDS). It has brought some of the semantics into the syntax.
Temporal logic programming
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

We can distinguish two views of logic, the declarative and the imperative. The declarative view is the traditional one, and it manifests itself both syntactically and semantically. Syntactically a logical system is taken as being characterized by its set of theorems. It is not important how these theorems are generated. Two different algorithmic systems generating the same set of theorems are considered as producing the same logic. Semantically a logic is considered as a set of formulae valid in all models.
Combining temporal logic system
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

This chapter is a continuation of the work started in Volume 1, Chapter 14 (which also appeared as [Finger and Gabbay, 1992]) on combining temporal logics. Here, four combination methods are described and studied with respect to the transfer of logical properties from the component one-dimensional temporal logics to the resulting combined two-dimensional temporal logic. Three basic logical properties are analysed, namely soundness, completeness and decidability. Meanwhile Gabbay’s [Gabbay, 1998) book Fibring Logics has appeared.
Extensional semantics
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

In Volume 1, and in some of the preceding chapters, we have seen how temporal logic can be used to reason about program behaviour. In describing a new programming language (or a new way of constructing reactive systems) we can often use temporal logic to specify the desired behaviour of any given program. Thus we are giving temporal semantics to the language. There are two main ways of approaching this task: intensionally or extensionally. Real programs will exhibit extensional actions, i.e. observable changes in values of program variables. However, specifiers may want to describe the behaviour in terms of states and transitions between them. These intensional states are abstract objects in their own right and are not necessarily determined by the values they associate with program variables. An example of this approach is the fair transitions system of [Manna and Pnueli, 1981] and [Pnueli, 1986] which generalizes such approaches as CSP and Petri nets. If such intensional states determine values for the observable variables, which they generally do, then it is straightforward to associate an extensional action--a change in variable values-with each transition between intensional states. However, the converse relationship is more problematic and there is not necessarily exactly one or indeed any one intensional transition associated with each extensional action.
Temporal Active Databases: A Background For Creating And Detecting Time Paradoxes
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

In this chapter we show how to create logical links between information associated to possibly distinct times in history. We can then discuss the effects that changes in history may have upon those temporal links.

For the purpose of establishing these links, we extend a valid-time database with temporal rules, and we provide those rules with an execution semantics; the resulting combined system is called an active valid-time database.
Semantical presentation of temporal connectives
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

In this chapter we discuss in more detail some of the aspects involved in presenting a temporal logic semantically. We deal with propositional temporal logics but the same applies to predicate temporal logics.
A general theory of axiomatization for propositional temporal logic
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

There are many possible connectives and therefore many possible logics. Some connectives are specially distinguished either because they appear in applications or because they represent natural technical features of flows of time. We have mentioned before the connectives Fq, Pq (‘It will be the case that q’ and ‘It was the case that q’) as natural connectives to consider. In fact, they should be included in any temporal logic. Other natural connectives are S(p,q) and U(p, q), and the technically important Stavi connectives S´(p,q) and U´(p,q). These are also important from the point of view of expressive power of temporal logic.
Expressive power of one-dimensional temporal connectives: basic concepts
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

This chapter deals with the basic concepts and theorems involved in measuring the expressive power of a temporal system. We deal here with one-dimensional temporal logics, those in which the unit for truth value is a point of the flow of time. Below we define the notions and connectives involved. These definitions are a special case of the definition in chapter 4. We repeat them here for the sake of clarity and relative independence of chapters.
H-dimension
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

We saw that expressive completeness of a set of connectives depends on the flow of time. Kamp’s Since and Until are expressively complete over the integers but not over the rationals. The natural question to ask is, given a flow of time, (T, <, =), can one find a finite set of connectives which is expressively complete over (T, <, =)?We shall see in section 13.2 below that a necessary condition for a positive answer to the question is that (T, <, =) has a property known as finite Henkin or H-dimension. We can use this to show that because some flows of time have infinite H-dimension, they admit no finite expressively complete set of connectives. The property of finite H-dimension is important. In sections 13.4 and 13.5 we give a method of generating more and more flows of time with finite H-dimension. Some of these flows are non-linear. In section 13.6 we examine a notion related to finite H-dimension, the so-called k-variable property.
Canonical Models and Filtration
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

In this chapter we consider two best known methods of obtaining completeness results. One of them-themethod of canonical models-givena consistent logicLin Extlnt or NExtK, constructs a canonical Kripke model 𝔐L= (𝔉L, 𝔉L) characterizingL.Sometimes the frame 𝔉Lturns out to be a frame forL,and then we can say at once thatLis Kripke complete. Another method, known asfiltration,is intended for establishing the finite approximability by means of extracting from the canonical models finite refutation frames.
Kripke Completeness
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

Perhaps the most desirable property of a logic is its decidability. However, the main tool for proving it (at least in the realm of modal and superintuitionistic logics) is Harrop’s theorem, according to which the decidability of a finitely axiomatizable logic follows from its finite approximability. Likewise, to prove many other properties of a logic, it is desirable first to establish its completeness with respect to some good class of frames, the simpler the better.
Finite Approximability
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

Let us now go one step down the hierarchy of frame classes and consider a stronger form of completeness, viz., completeness with respect to the class of finite frames or, in other terms, finite approximability. We have already met with one way of proving this property—the filtration method, requiring a specialad hoctechnique in each particular case. Now we will show a few other methods which provide us in fact with general syntactical and semantic sufficient conditions of finite approximability.
Tabularity
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

Now we consider tabular and locally tabular modal and superintuitionistic logics. The main question we try to answer here is how to determine whether a given logic is tabular or locally tabular. First we establish that every tabular modal (no matter normal or not) and silogic is finitely axiomatizable. This will be done with the help of a syntactic criterion of tabularity which uses the following formulas:
Interpolation
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

Recall that a logicLis said to have theCraig interpolation propertyif, for every implicationα鄤βinL,there exists a formula γ, called aninterpolantforα鄤βinL,such thatα鄤,γ∈L,γ鄤β ∈and Var γ ⊆,;Varaα  ∩ Varβ. In this chapter we present the most important semantic methods of proving and disproving the interpolation property of modal and superintuitionistic logics.
The Decidability of Logics
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

The first and perhaps most important algorithmic question arising immediately after creating a logic is the question of its decidability: is there an algorithm which is c;apable of deciding, given an arbitrary formula, whether it belongs to the logic or not?
Do it yourself
Authors: Clark Malcolm
Published: November 2023

The handful of commands which were introduced in the last chapter were introduced with the idea that we can now re-run the EXAMPLE. Tex file, changing the various parameters. We can generate more white space between lines by increasing the value associated with the baseline skip, put more white space between paragraphs, change the font we are using; but more useful at this stage would be to alter the page dimensions. Changing the page width in particular will most quickly generate interesting, educational, results.
Beginning mathematics
Authors: Clark Malcolm
Published: November 2023

In many respects this is where Tex gets to be lots more fun. There is something extraordinarily satisfying about being able to set mathematical formulae. Tex permits this both within the text ( text style) and displayed on lines by themselves (display style). The same expression can be given in text or display style, and there will be a few stylistic changes, which Tex takes care of automatically.
More words
Authors: Clark Malcolm
Published: November 2023

Even the most erudite and involved mathematical treatise needs the glue of words to hold the pieces together. Despite the incredible ease and simplicity with which ‘!EX can handle equations, as we have already discovered, it also handles non-technical material in a straightforward and direct way. Unfortunately, the rules for ‘normal’ text are less rigid. Why unfortunately? The greater the amount of choice, the greater the amount of disagreement. We can generally agree when equations ‘look wrong’, but decid ing when text ‘looks right’ is rather more tricky. The general approach adopted here is to concentrate on the plainer aspects, assuming that the content is the important component, and that the presentation should be as bland and anonymous as possible. At the extreme, we hope the presentation will be invisible. If we are very fortunate, it may even contribute subtly to the message. But, before the heavyweight material, something carefree.
Commands
Authors: Clark Malcolm
Published: November 2023

A great many of ‘IE;X’s mysteries are bound up in commands. Alternative terms are ‘macros’, ‘control sequences’, or ‘definitions’: although Knuth refers to them as macros, unless you are familiar with computer science jargon, there is no advantage in this usage. The 300 or so fundamental ‘Tex commands are the building blocks which can be assembled quite easily (and all too readily) into rather arcane powerful commands. It is easy to be overwhelmed by the apparent simplicity and sheer power of commands. It is this feature of ‘Tex which gives it the appearance of a programming language, rather than a simple set of commands. One of the first places we might look to see working commands could be plain. tex which is the record of all that plain ‘Tex.
Boxing
Authors: Clark Malcolm
Published: November 2023

To explain how Tex handles boxes we can first return briefly to some of the ideas introduced very briefly in Chapter 3. There the boxes considered were those which were associated with individual letters which were themselves assembled into a larger box - the line. The horizontal lines are then stacked together into a vertical list of boxes. Let’s go back to the simplest cases. There are two fundamental sorts of boxes: horizontal and vertical. When Tex is creating a line of text, it is creating a horizontal box. When it assembles several paragraphs into a page, it is creating a vertical box.
Coininands#1
Authors: Clark Malcolm
Published: November 2023

Commands (definition or macro) start to get much more interesting when they have parameters. The basic form of the definition of a command with parameters is something like where the first #1 says that there is to be one single parameter. The repetition of #1 within the braces merely tells ‘It;X how you want the parameter (or argument) to be treated. Having defined\one, we can use it by saying \one{argument). All this does is to place argument into the text at this point. In this instance the command is dull and rather pointless, so let’s have a look at a more interesting single-parameter command.
Matrix Manipulations
Authors: Clark Malcolm
Published: November 2023

So far we have merely looked at fairly simple equations which can conveniently be written either in the text, or in a one-line display. We have not considered more ‘two-dimensional’ structures, like matrices. It is probably fair to say that large equations, which go over more than one line, as well as matrices (or tables) are unlikely to be found in text rather than display. Almost all of the following discussion therefore assumes implicitly that we are talking about displayed equations and formulae.
Pages
Authors: Clark Malcolm
Published: November 2023

Although a great deal of Tex has been presented already, only a very small amount has been directed towards the handling of fairly ordinary text. Even the most esoteric mathematics usually requires the ‘glue’ of words to hold it together. In this chapter, some of the tools and techniques available will be discussed. Text looks deceptively simple. It is probably considerably more difficult to format text than mathematics. The range of possibilities for text is so much greater. Examination of a couple of non-technical books will probably yield more variation in typesetting than com parable technical (especially mathematical) books.