bluuebunny/crossref_metadata_2025 huggingface dataset:
Stratification theorems
Authors: Evens Leonard
Published: November 2023

Let k be a field of characteristic p, G a finite group, and E an elementary abelian p-subgroup of G. We shall identify Eˆ= Hom(E, F  p) with Im{H  1(E, F  p) → H  1(E, k)} for p = 2 or Im{H  2(E, Z) → H  2(E, k)} for p odd. (For p odd, Eˆ may also be thought of as the image of the composite map [ineq] Similarly, we shall identify E* = Hom(E, k) with the k-subspace of H(E) generated by Eˆ: (This is a subspace of degree 1 for p = 2 and degree 2 for p > 2.) For p = 2, H(E) = H*(E, k) ≅ S(E*) (the symmetric algebra on E*). For p > 2, H(E) = S(E*) ⊕ J where J is the nilpotent ideal generated by H  1(E, k)2 ⊂ H  2(E, k); in fact, J = rad H(E).
Summability methods in general
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

A summability method for a series [ineq] is simply any scheme for as- signing to it a unique number which we then call the ‘sum’ of the series or the ‘limit’ of the associated sequence of partial sums {sn}, In practice, working with a method can usually be reduced to consideration of one of the following transformations.
Borel’s methods of summability
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

Although the basic definitions have been given in the previous chapter, we shall, for completeness, repeat them here. We shall also introduce some additional notation.
Abelian theorems
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

As stated in chapter 5, certain results are known as Abelian theorems because they are of the nature of Abel’s theorem in the limit of a power series at its radius of convergence.
Applications of Borel’s methods
Authors: Shawyer Bruce , Watson Bruce
Published: November 2023

The bibliography of articles that can be classed as applications of Borel’s methods has about 70 entries. Of these, only one predates 1970. That is the article by Cunningham in 1905 A[18] entitled An extension of Borel’s exponential method of summation of divergent series applied to linear differential equations. Here, Cunningham’s object was to make more precise the connection between Laplace’s integral solutions of linear differential equations with rational coefficients and of rank unity in the neighbourhood of z = ∞, and the power series solution with an exponential factor. He noted that Poincaré had developed asymptotic expansions of the integrals, and had found them useful for numerical calculations. However, these asymptotic expansions were of little value in that they did not represent unique analytic functions.
Zariski Categories
Authors: Yves Diers
Published: November 2023

We use the axiomatic method, which deals with categories equipped with a specified structure rather than categories described concretely by their objects and their morphisms. The so-called Zariski categories have such a structure, and we claim here that it is the basic structure of categories of commutative algebras. We shall prove the validity of this claim by developing, in an arbitrary Zariski category, elementary commutative algebra and algebraic geometry. The structure of Zariski categories relies on the notion of codisjunctors. This universal construction is introduced in order to describe the calculus of fractions and is used extensively. In Zariski categories, the congruences, also called effective equivalence relations, play the important role of ideals. A kind of De Morgan law for codisjunctors of congruences is assumed. Among the basic properties of Zariski categories are the flatness properties. We use the following definition: a morphism f:  A → B is flat in A if the pushout functor A/ A→ B/A along f preserves monomorphisms. Codisjunctors are assumed to be flat morphisms.
Review of stability theory
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

The earliest results in the theory of stability of smooth mappings are due to Whitney. The case of maps ℝn →with ℝp  p 2 ≥2n -1 is elucidated in Whitney (1943) and the case p = n = 2 in Whitney (1955). In Whitney (1958) these and other results are summarized, possible generalizations discussed, and the general conjecture formulated that a generic map has all its germs c ∞=-stable; also, this is placed in a function space setting.
The worst case setting: general results
Authors: Werschulz Arthur G
Published: November 2023

In this chapter, we describe general results for the worst case setting. Recall that we are given a set F of problem elements, as well as a normed linear space G with norm || · ||. We wish to approximate the solution operator S: F → G. We measure error and cost in a worst case sense.
Other problems in the worst case setting
Authors: Werschulz Arthur G
Published: November 2023

In the previous chapter, we applied the techniques of information-based complexity to find algorithms that are optimal in the worst case setting for elliptic partial differential equations (PDE). In this chapter, we apply these techniques to find algorithms that are (worst case) optimal for several other important classes of problems. The first section deals with elliptic systems of PDE. The second section deals with well-posed integral equations (IE), namely the Fredholm problem of the second kind. The third section is a general treatment of linear ill-posed problems, with special emphasis on the Fredholm IE of the first kind. Finally, the fourth section deals with ordinary differential equations (ODE).
Functional Spaces in Viscoplasticity
Authors: Ionescu Ioan R , Sofonea Mircea
Published: November 2023

The purpose of this chapter is to provide some fundamental results on functional spaces, as used in viscoplasticity, for later reference. These spaces are various spaces of distributions associated with the deformation operator and the divergence operator, or spaces of vector distributions. To restrict the chapter to a reasonable length most of the basic theorems are stated without proofs. However, for the convenience of the reader some useful properties of the spaces related to the divergence operator are presented here with proofs (Section 2.3). For other results which can be easily found in text books or monographs adequate references are given. The notations stated in this chapter are used all over the book.
Introduction
Authors: Robinson Derek W
Published: November 2023

The purpose of this book is to develop the basic theory of elliptic opera tors on Lie groups and thereby extend the conventional theory of parabolic evolution equations to a natural non-commutative context. The use of a Lie group as the underlying manifold introduces many complications not encountered on Euclidean space. The usual partial differential operators are replaced by representatives of the corresponding Lie algebra and this leads to a variety of difficulties even in the description of operators with constant coefficients. The Lie group theory of strongly elliptic operators with constant coefficients is comparable in complexity to the Euclidean theory with variable coefficients. Locally the two theories are in fact equiv alent and this is the key to the construction of the Lie group theory. The basic existence, regularity, and boundedness, properties are established by means of local Euclidean estimates.
Analytic Elements
Authors: Robinson Derek W
Published: November 2023

There are two aspects of the estimates (2.1) of Theorem 2.2 which are of importance. First, the growth with n of the Cn-norm of S allowed the foregoing comparison of the analytic elements of the elliptic operators and the representation.
On totally irregular simple collineation groups
Authors: Gorn;alves C Y Hoand A
Published: November 2023

Classifications of Desarguesian planes by doubly transitive collineation groups (Ostrom-Wagner) or by the property that either each point is a centre or each line is an axis of a perspectivity (Cofman Piper) are examples of totally irregular collineation groups with few orbits. Cyclic planes (in general planar difference sets) also admit totally irregular collineation groups, by using Hall’s multiplier theorem (Hughes and Piper (1973)).
Stabilisers of hyperovals in PG(2, 32)
Authors: O’Keefe Christine M , Penttila Tim , Praeger Cheryl E
Published: November 2023

We shall be concerned with the collineation stabiliser G and the homography stabiliser H of a hyperoval ℋ in PG(2, q), that is, the stabilisers of 5/ in the collineation group PTL(3, q) and homography group PGL(3, q) of PG(2, q) respectively.
Applications of the p-Adic Subspace Theorem
Authors: Everest G R
Published: November 2023

Given an integer m⩾2 and rational numbers A1,…, Am, u1,…, Um not all zero, we obtain a rational sequence using the linear recurrence relation, [eq] Sequences of this kind have been studied extensively and the theory has a long and colourful history. One of the earliest and best known is the Fibonacci sequence. This begins u1 = u2 = 1, and continues via the recurrence relation A slightly more exotic example occurs with the problem of the man who walks along an infinite ladder.
Introduction
Authors: Klingen Norbert
Published: November 2023

As prerequisites for reading this book we assume elementary facts from algebra and group theory including ordinary representation theory. A concise introduction to the fundamental concepts needed from algebraic number theory, for example prime decomposition, zeta function, density theorems, Artin L-functions, is given in Chapter I. This chapter already contains the key link between prime decomposition and group theory.
Arithmetical homomorphisms
Authors: Klingen Norbert
Published: November 2023

From Chapter III, theorem (1.4) we know that arithmetically equivalent fields have lots of invariants in common; in particular their zeta functions agree. From this it followed that the products hR of class number and regulator coincide in arithmetically equivalent fields (III.1.4 j)), but the relation between the class numbers themselves remained open. So one important subject in this chapter is the investigation of class numbers.
Kroneckerian fields
Authors: Klingen Norbert
Published: November 2023

In this chapter we study fields K for which any Kronecker equivalent field K′ must already be conjugate to K. We call such fields Kroneckerian since Kronecker’s concept [64] of characterizing a field in terms of its decomposition law is established for these fields in the strongest possible form.
Connections
Authors: Petrie Ted , Randall John
Published: November 2023

In this chapter, we introduce ideas which are fundamental to what follows. First we define a connection on a vector bundle E over a differentiable manifold M, and give some basic results.
The Virtual Moduli Space M Of Virtual Self-Dual Connections
Authors: Petrie Ted , Randall John
Published: November 2023

In the reducible case it is easy to verify this assertion and in fact to see that the isomorphism is r(dE)-equivariant for any SO(3) virtual connection on E.
Intersection Forms On 4-Manifolds
Authors: Petrie Ted , Randall John
Published: November 2023

In this chapter, we bring together all of the results we have previously discussed to prove results about the intersection forms oh differentiable 4-manifolds. Our aim is to use a cobordism argument to show that certain intersection forms cannot arise.
Moduli Space For Invariant Self-Dual Virtual Connections
Authors: Petrie Ted , Randall John
Published: November 2023

Let G be a compact Lie group. Previously (Section 2.8) the term G-vector bundle meant a vector bundle with group G as the structure group. It has another accepted meaning (used throughout the remainder of this text) when G acts on the base space of the vector bundle. In this case there are two groups to deal with, G and the structure group of the bundle. We shall deal exclusively with the structure groups SO(2) and SO(3). In this context, a G-vector bundle is a vector bundle (of dimension 2 or 3) with G acting to preserve the inner product and orientation, commuting with base projection, and mapping fibres directly to fibres (see [1]).
An overview of SHEEP
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

Unlike the other computer algebra systems described in this series of volumes, SHEEP is a system constructed specifically for the task of doing lengthy algebraic manipulations of the kind encountered in general relativity.
Using CLASSI to classify metrics
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

The CLASSI package (Åman 1987) is introduced in this chapter by an account of the theoretical basis of the classification procedures which it embodies. Then the direct use of the classification programs is described, followed by further details of extra facilities such as the handling of complex quantities. The form of the results and the future directions of work in this area are considered. Finally a list of all CLASS! modules, including those not directly related to the classification algorithms, is given.
Stensor
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

In fact, just as with SHEEP or CLASS!, the concept of tensor is not fully understood by the system in the sense that the ‘‘tensors” need not obey tensor transformation laws: thus any indexed (or scalar) object can be considered a tensor in STENSOR.
Introduction
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

The lecture notes that follow are aimed at those who have already done a first course in general relativity and have acquired an elementary knowledge of programming in the algebraic mode of REDUCE 3.3 (Hearn 1987). The intention is not to present the reader with a software package that will function as a sort of black box which processes input and churns out output. The reader is rather invited to work through the details of the program given here and then to go and write his/her own program, at least at the level of the algebraic mode which is all that will concern us here. In several places in the text, reference is made to a library of files called GRLIB. These are a set of straightforward REDUCE programs for computing the more commonly used geometrical objects in general relativity.
Geometrical preliminaries
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

In order to establish an agreed notation we begin with a brief summary of some of the geometrical concepts that arise in theories of gravitation and, in particular, in Einstein’s theory.
Calculations on coordinate bases
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

The golden rule in performing general relativistic calculations is to use, whenever possible, anholonomic frames for which the metric components are constants. The reason for this is that the expressions one obtains for the components of the various tensors that arise are normally simpler in an anholonomic frame. However sometimes it is necessary or convenient to calculate with a straightforward holonomic or coordinate frame and it is to this type of computation that we first turn our attention in this section. A typical requirement would be, for instance, to compute the Riemann, Ricci or Einstein tensors for a given space time metric, or to look for the Killing vector fields of the space-time. If there are also electromagnetic fields present the Einstein-Maxwell equations will have to be computed or perhaps one may wish to apply the Rainich (1925) conditions.
Introduction: overview of MAPLE applications to general relativity
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

For the past thirty years researchers have been using computer algebra systems to perform some of the lengthy and tedious computations involved in research in general relativity. The pioneers of the subject Fletcher et al. (1967), d’lnverno (1969), Barton and Fitch (1972), and Pavelle (1975) mainly used the systems to compute the Christoffel symbols and the components of the curvature tensor, Weyl tensor, Ricci tensor and the Einstein tensor from the components of a given space time metric in the natural basis. The main purpose was to verify that the metric actually satisfies Einstein’s field equations. A further early application was the determination of the Petrov type for all the Harrison’s (1959) solutions by d’Inverno and Russell-Clark (1971). Programs were also written by Ernst (1968) and Harrison (1969) for the computation of the connection coefficients and curvature components, corresponding to a given rigid null or orthonormal frame using Cartan’s structural equations.
The Petrov classification of the Weyl tensor
Authors: MacCallum M A H , Skea J E F , McLenaghan R G , McCrea J D
Published: November 2023

The determination of the Petrov type of the Weyl tensor for a given space-time metric is important in exact solution theory, in particular in the equivalence problem where it forms an important part in the algorithm for determining whether two space-time metrics are locally isometric. The determination of the Segre type of the Ricci tensor (not discussed here) is also an important part of the algorithm. It should be noted that most Ricci tensor classification algorithms (see Joly and MacCallum 1990) require the determination of the Petrov type of the Plebanski tensor. Thus the determination of the Petrov type of tensor with Weyl tensor symmetries is seen to be particularly fundamental especially in the context of the equivalence problem.

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Analysis of lifetimes with covariates
Authors: Ansell J I , Phillips M J
Published: November 2023

In many reliability studies as well as the information on the component’s (or system’s) lifetime there is also available other information about the component (system) or its environment. Typically there may be information on the design of the component or the wear the component has suffered. Hence instead of just recording a set of lifetimes there may be associated with each lifetime other variables. These variables are known as covariates. (The term is often used to cover factors, see Section 3.2, as well as measurable variables.) This extra information should yield more understanding about the performance of the component or system. The more information used the better the understanding, hopefully. Recently there has been considerable interest in the analysis of such data sets in the reliability and related literature. Part of the interest may be due to the availability of the proportional hazards model, suggested by Cox (1972, 1975). This has focused interest in relating both variables and factors to components’ lifetimes. However, the proportional hazards model is not the only model available and a number of approaches predate it, such as the use of the accelerated failure time model (Nelson, 1993) and the Weibull regression model (Smith, 1991). All the models and techniques discussed in this chapter are regression-type techniques.
Practical aspects of reliability data analysis
Authors: Ansell J I , Phillips M J
Published: November 2023

Previous chapters have explored models and techniques which provide the basis for reliability data analysis. In this chapter the practical aspects are discussed. Before starting on such a discussion, though, it is worth re marking that many reliability analyses do not require the use of statistical techniques. Even for those which do, the use is often very limited and routine. Sometimes their use may even be misguided, see O’Connor (1991). The statistical analysis will often not play the central role in the full analysis and will mainly be used to support the conclusions about the main objective. However, the use of statistical analysis can provide considerable information to enable analysts to make decisions. Often the value of this analysis is overlooked and the full power of the statistical tools available underutilized. One reason is that in the past reliability engineers have seen sophisticated techniques applied to their data with relatively little advantage to them. There has been a concentration on the technique rather than the underlying problem. When statistical methods are used the approach should be problem-led rather than technique-led. This means that the objective of the analysis should always be to the fore in the study. The objective is the key to the analysis. The way to achieve the objective is through a strategy. The first section of this chapter will there- fore concentrate on objectives and strategies. The strategy offered in this chapter is not prescriptive; hence unlike other authors a flowchart is not provided for the possible stages in the analysis. Obviously different objectives will have specific strategies developed for them.
Topological homology
Authors: Eschmeier Jörg , Putinar Mihai
Published: November 2023

The homology of topological algebras and of topological modules over topological algebras has been developed relatively late and rather at a tangent to the main directions of research in homological algebra. However, this gap has been considerably reduced in recent times, when the interest in the Hochschild cohomology of operator algebras has increased, partly due to the elaboration of cyclic cohomology and of non-commutative geometries.
The first limit problem in MPT
Authors: Grenander Ulf
Published: November 2023

In this and the following two chapters on metric pattern theory (MPT) we shall study probabilistic limit theorems on regular structures. They are related to the limit theorems or standard algebraic structures such as semigroups, groups, vector spaces and algebras (Notes 9.A) but differ from them in that the algebraic properties of regular structures, as studied in Part I, are quite different. Instead of ‘addition’, ‘multiplication’ by scalars, or whatever the unary and binary operations are called in standard situations, we now have the laws of local interaction, action of similarity transformations and coupling connectors.
Continuum-based models
Authors: Grenander Ulf
Published: November 2023

In this first section we shall study picture ensembles consisting of human hands captured by a visible light digital camera under various degrees of optical degradation. Hand shapes are very characteristic but also exhibit a lot of variability, so that inferences take place in a high-or infinite-dimensional parameter space – abstract inference.We aim for both intrinsic image understanding (see Chapter 13) in the form of image restoration and objective landmark determination and for extrinsic understanding, here meaning abnormality detection.
Hamilton-Jacobi Theory
Authors: Woodhouse N M J
Published: November 2023

Eqn (4.1.1) is the Hamilton-Jacobi equation (more accurately, it is the time-independent form of the Hamilton-Jacobi equation). Although it was originally introduced as a device for obtaining analytic solutions to mechanical problems, it is, in fact, of very little use in that context since the only generally applicable method for solving such first-order partial differential equations involves the reverse construction and depends, as a first step, on the integration of Hamilton’s equations (for example, see Courant and Hilbert 1962). Apart from this the only other practical method is to separate the variables, and this can only be done for a very special class of Hamiltonians.
Classical Fields
Authors: Woodhouse N M J
Published: November 2023

At a formal level, much of the symplectic geometry that underlies the Hamiltonian formulation of classical mechanics extends to field theories, and can be used as the starting point for quantization. In this chapter, we shall look at relativistic field theories, and at the connection between the Lagrangian density and the symplectic structure on the space of solutions.1 To prepare the ground for quantization, we shall look at how the splitting of the fields into their positive and negative frequency parts can be interpreted as a complex polarization in the classical phase space.
Circular words
Authors: Reutenauer Christophe
Published: November 2023

There are many links between the free Lie algebra and circular words; the most immediate is the equality of the homogeneous dimension of the former, given by the Witt formula, and of the number of primitive necklaces. The aim of this chapter is to study circular words as an end in itself.
Discrete Wavelet Transform
Authors: Freeden W , Gervens T , Schreiner M
Published: November 2023

In the wavelet reconstruction formulae studied so far, the discretization of the corresponding wavelet transforms must be made directly by computing the relevant integrals for all the necessary values of rotation parameters. In Euclidean theory, however, a multiresolution analysis for performing discrete wavelet analysis and synthesis is known, which is completely recursive and, therefore, ideal for computations. This form of multiresolution analysis is certainly one of the reasons why the wavelet theory has gained such widespread popularity. In what follows we make the attempt to generalize as much as possible from this multiresolution procedure to the spherical context.
Open problems
Authors: Baaz Matthias
Published: November 2023

1 Bounded Arithmetic

The study of theories of bounded arithmetic was largely begun by work of R. Parikh [51] and the ground-breaking work of J. Paris and A. Wilkie [54, 73, 58, 57, 72, 25]. Especially with the work of S. Buss [10], proof theoretic problems of weak theories of bounded arithmetic were seen to be connected with problems of computational complexity theory.
0 Introduction
Authors: Gurtin Morton E
Published: November 2023

In this treatise I discuss the dynamics of two-phase systems within the framework of modern continuum thermodynamics, assuming throughout a sharp interface separating bulk phases. I consider only non-deformable continua, neglecting mass transport, and to avoid the geometrical complications associated with the motion of surfaces in ℝ, I restrict attention to evolution in the plane. I consider two general theories: a purely mechanical theory that leads to a generalization of the classical curve-shortening equation
Regularized Theory for Smooth Unstable Energies; Dependence of Interfacial Energy
Authors: Gurtin Morton E
Published: November 2023

Unstable interfacial energies lead to backward-parabolic evolution equations for the interface. In this section we regularize the evolution equations by allowing for a curvature-dependent interfacial energy, a dependence that requires the introduction of capillary moments in conjunction with a more general balance law for moments. (a) Balance of forces and moments; power Consider an evolving interface 𝓈 with r a corresponding parametrization. The micromechanics of the interface is described by four interfacial fields: the capillary force C(s, t), the interactive force B(s, t), and the interactive moment m(s, t), discussed in Section 4.1, and an additional field, the capillary moment M(s, t), which represents the moment within the interface exerted across s.
Node-cell mappings for the cell vertex method
Authors: Morton K W
Published: November 2023

In the cell centre finite volume method, as in the Galerkin finite element method, there is no problem in identifying each discrete equation with a corresponding unknown. Hence it is a straight-forward matter to set up an iteration procedure to solve the equations, based for example on Runge-Kutta time-stepping schemes. The difficulties arise in establishing the stability of the equations and ensuring the convergence of the iteration at a reasonable rate.
Mesh adaption for compressible flows
Authors: Palmerio Bernadette
Published: November 2023

One of the principles of numerical analysis is to compute the solution of approximate problems, instead of the solution of continuous ones. The accordance between the discrete solution and the continuous one results from various, often antagonist, options such as choosing a fine mesh or saving computer cost. In most cases, the discrete solution is constructed in two phases: firstly a mesh is generated from the point of view of taking into account geometrical data, and then the solution is computed on the mesh, from the physical data, by solving a differential system.
The construction and behaviour of some unstructured grid algorithms for compressible flows
Authors: Peraire J , Vahdati M , Peiró J
Published: November 2023

The methods currently available for the solution of the equations of compressible flow have traditionally been classified as being of either centered or upwind type. Centered schemes are generally developed from the use of central difference type approximations to derivative operators [1], [2], [3], [4], [5]. Current upwind schemes, on the other hand, employ the solution of a one dimensional Riemann problem in the construction of the discrete form of the governing equations and are computationally more expensive [6], [7], [8], [9], [10]. Frequently, centered schemes rely on the explicit addition of artificial viscosity and this requires the empirical determination of the corresponding free parameters. Upwind methods are generally regarded as being parameter free.
A dynamically adaptive Godunov scheme for reservoir simulation on large aspect ratio grids
Authors: Edwards Michael G
Published: November 2023

1 Introduction

Large aspect ratio grids are commonly used in the simulation of a vertical cross section of a reservoir, where height is much smaller than length. Unit aspect ratio grids are typical for areal simulations.

The combination of a higher order Godunov scheme with dynamic grid adaptivity applied to reservoir simulation is significantly faster than the standard simulator scheme (first order upwind on a uniform grid) while producing far superior results for areal problems (Edwards 1992).

The results obtained from applying the adaptive higher order Godunov scheme to a vertical cross section using a grid with cells of large aspect ratio show a deterioration in accuracy as the aspect ratio is increased. Large local errors appear in the solution at grid interfaces which separate cells of different levels.
Shock capturing using auto-adaptive finite elements
Authors: Richter R , Leyland P
Published: November 2023

Shock capturing properties via numerical or artificial dissipation is highly linked to the underlying grid structure, and can be improved by using local refinement on adapted grids. Grid alignment is even necessary when one dimensional solvers are adapted to multidimensional problems, as is the case of many modern TVD methods. Also, complex flow calculations for solving secondary shock formation in shock-shock interactions are particularly difficult to simulate accurately. By the use of mesh adaptivity, these effects can be more accurately illustrated within the numerical simulation. Here, combining the relative grid dependency of the numerical scheme and auto-adaptive grids using refinement-derefinement algorithms on unstructured meshes provides a robust tool for capturing such shock formations, and a generic remeshing strategy for unsteady flows. Combining such techniques with directional scaling and stretching, using an elliptic spring analogy, further increases the shock precision, and leads to solutions of almost “shock fitted” quality. Unstructured meshes allow more freedom for these kinds of adaptivity algorithms than structured grids; the latter requiring complicated embedding techniques to achieve similar results.
A cartesian grid scheme for gas dynamic flows that involve complex geometries
Authors: Quirk James J
Published: November 2023

The desire to simulate flows around geometrically complex bodies has led many members of the shock capturing community to abandon schemes which employ body-fitted grids in favour of schemes which employ unstructured meshes. Now such is the success of unstructured grids, there is a danger that they will become de rigueur. Given the ever increasing reliance placed on computational results, such a state of affairs would give cause for some concern. If nothing else, the exercise of using several computer codes to cross check numerical results becomes ill-founded if all codes follow the same methodology. In this paper we outline an alternative approach for dealing with complex two-dimensional geometries, the so-called cartesian boundary method; solid bodies blank out areas of a background cartesian mesh and the cut cells which arise along solid boundaries are singled out for special treatment during the integration of the discretized flow solution.
A new implicit second order scheme based on a kinetic interpretation for solving the Euler Equations
Authors: Vuyst Florian De
Published: November 2023

Over the last decade, several numerical schemes for solving the compressible Euler equations have been built by the way of the kinetic interpretation of these equations. Variants of these Boltzmann-type schemes have been studied by Deshpande, Kaniel and others, using a physical repartition of the particle velocity: the Maxwellian distribution exp(-lvl2 /2). But the main drawback of these methods is their important numerical diffusion. In Perthame (1990) a new class of Boltzmann type schemes was introduced by considering compactly supported velocity distribution functions. Techniques for achieving second order accuracy have been elaborated in Perthame & Qiu (1991), guaranteeing the preservation of the positivity of density and pressure in one or two space variables. The underlying kinetic interpretation also allows the construction of fully multidimensional schemes taking into account the corner effects, and the treatment of wall boundary conditions by the way of specular reflection. In this paper, we present a classical two-dimensional finite volume method which uses the numerical fluxes coming from the one-dimensional scheme. Second order accuracy is performed by a generalized Van Leer’s MUSCL method, applicable to structured or unstructured meshes. We also extend these schemes to the simulation of high speed flows, where the nonequilibrium effects have to be dealt with.
A comparison of several implicit smoothing methods
Authors: Blazek J , Krollm N , Rossow C C
Published: November 2023

In recent years, several different numerical methods for the solution of Euler and Navier-Stokes equations were introduced in the literature. Due to their simplicity the explicit multistage time stepping schemes originated by Jameson et al (1981) quickly gained popularity and today they are in wide-spread use. In combination with various convergence acceleration techniques, such as local time stepping, residual smoothing and in particular multigrid, the multistage schemes proved to be very efficient for the calculation of both inviscid and viscous steady flow fields (Radespiel et al 1990 and Radespiel et al 1991). The use of implicit residual smoothing (IRS) extends the stability region of the basic explicit time stepping scheme and increases thereby the maximum allowable Courant number. Theoretically no restrictions on the time step are imposed provided that a large enough smoothing coefficient is used. In addition the residual smoothing strongly effects the damping properties of the scheme and the particular technique used may lead to a considerably improved damping of the high frequency variations of the residuals. This is essential for the design of fast and robust multigrid schemes. The drawback of the commonly used centrally weighted IRS is the restriction of the smoothing coefficient to only small values in practical computations. In this way only a moderate raise of the CFL number is possible. Recently, upwind biased IRS operators have been developed by the authors (Blazek et al. 1991) which for one-dimensional flows allow considerably higher CFL numbers.
Implementation of non-reflecting boundary conditions in an unsteady flow simulation system
Authors: Engel K , Faden M , Pokorny S
Published: November 2023

An interactive flow simulation system is under development by the authors to study and analyze the unsteady flow phenomena in turbomachines. Special emphasis is put on the unsteady effects of the flow through the blades of counter-rotating ducted fan rotors. The whole system has been setup on a parallel computer which provides the necessary computational power together with the required interactive access.
Valuation Systems and Consequence Relations
Authors: Ryan Mark , Sadler Martin
Published: November 2023

This chapter is an introduction to some of the basic concepts and machinery of logic, concentrating on the contemporary uses of logic in computer science. It is not comprehensive, since many topics (such as equational reasoning, algebras, categories and computability) are introduced in other chapters. In particular, we concentrate on the applications of logics as reasoning systems about computing rather than on the foundational aspects of computer science which logic addresses. This is not a standard introduction to classical first order logic and its model theory, such treatments being widely available elsewhere [Hamilton, 1978; Hodges, 1983; Makowsky, ]. Rather, we attempt to bring together some topics from the logics currently used for reasoning about programs and about computer systems, and see them within a unifying framework at an appropriate level of abstraction.
Uncertainty Logics
Authors: Kyburg Henry E
Published: November 2023

The formalization of ordinary deductive inference is not uncontroversial. There are questions of modality, many-valued truth functions, and the like. But there is a solid core of agreement underlying the variations. This is not true of uncertain inference. Much, if not most, of our ordinary reasoning is forced to take account of uncertainty. That is, in ordinary reasoning and argument, either our premises are less than incorrigibly certain, or the rule of inference that we follow is less than valid in the classical sense—it may be possible for our premises to be true and our conclusion false. But there is very little agreement, as yet, on the ways in which uncertainty should be represented, or on the forms that inference involving uncertainty should take, or on the logic characterizing uncertain inference.
Temporal logic and automata
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

This chapter1 covers certain automata, their expressive power, non-deterministic versus deterministic automata, and their relation to games. We apply these results to prove the decidability of certain related monadic second-order logics (‘SIS’ and ‘S2S’). These logics are equivalent to some temporal logics, and also very useful in proving decidability of a host of other temporal logics, as we saw in Chapter 15 of Volume 1 of the current work.
Intervals and planning
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

As described in [van Benthem, 1995], there are many and various motivations for using an interval temporal logic, which include philosophical considerations of time and events, natural language, processes in computations and planning problems. In this chapter we will motivate the study of interval temporal logics by glancing at philosophical arguments, seeing some of the interesting mathematics which underlies questions of expressiveness, decidability and axiomatic completeness for such logics and considering their application to planning problems. We will also see reasons why interval temporal logics have not become very popular: these include undecidability and a lack of expressive completeness.
Many-dimensional system and generalized quantifiers
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

This chapter will study in more detail the properties of many-dimensional systems based on tree flows of time. As an application, we shall interpret generalized quantifiers in such systems.
The Declarative Past And Imperative Future
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

We propose a new paradigm in executable logic: that of the declarative past and imperative future. A future statement of temporal logic can be understood in two ways: the declarative way, that of describing the future as a temporal extension; and the imperative way, that of making sure that the future will happen the way we want. Since the future has not yet happened, we have a language which can be both declarative and imperative. We regard our theme as a natural meeting between the imperative and declarative paradigms.
A logical view of temporal databases
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

Traditional approaches to temporal databases are either not formal or deal only with static features, considering mainly a single state of the database. The temporal nature of data in temporal databases, however, requires a dynamic treatment of several features which are absent from non-temporal systems.
Calendar Logic
Authors: Gabbay Dov M , Reynolds Mark A , Finger Marcelo
Published: November 2023

In this chapter we link temporal logic to real-time calendar systems. Instead of the very general operators like ‘sometime in the future’ or ‘always in the future’, we introduce relativized operators of the kind ‘sometime next week’ or ‘always in office hours’. Calendar logic is developed in three stages. We start with a system for specifying everyday temporal notions like ‘hours’, ‘weeks’, ‘weekend’, ‘office hour’, ‘holiday’ within a given calendar system. In this system one can check whether a given point in time lies within an interval specified by a time term, or whether a subsumption relation holds between two time terms.
Temporal logics presented in the predicate calculus: language—metalanguage
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

Given a system of databases {Δt ⊢t), t ∈ T, we say in chapter 1 that there are two ways of describing the logical connections between the theories (Δt, ⊢t). One is to talk about the system declaratively from the outside, using a metalanguage M, and the other is to enrich the language of the theories At, add connectives and formulae to theory Δt, enabling it to express the way in which it is related to other theories.
Expressive completeness of Since and Until over integer and real time
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark
Published: November 2023

In this chapter we will show how the separation theorem — theorem 9.3.1 — can be used to prove the expressive completeness of the language with Until and Since over some very important flows of time. To do this we need to show the separation of the language over the flows. First, in section 10.2, we examine the discrete Dedekind complete flows of time, concentrating on the integers. This serves as a simple introduction to the techniques which we then use in section 10.3 to show separation of this language over any class of Dedekind complete flows of time. The separation procedure is performed in two stages: first some basic equivalences, called eliminations, are shown; then we show how to put eliminations together in the right order to achieve separation. In this section we prove that the language with the connectives S and U has the separation property over the flow of time of the integers. This proof appeared in [Gabbay, 1989a]. It can be seen that the proof will also work for the natural numbers.
From Logics to Classes of Logics
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

We have already met with sufficiently many concrete logics to make some generalizations. Instead of proving the same sort of theorems for each logic separately, we can consider big classes of logics and try to develop general methods for investigating their propertiesen masse.In this chapter we introduce rather abstract concepts of superintuitionistic and modal logics and discuss the general settings of problems associated with them to be examined in the rest of the book.
The Decidability of Logics’ Properties
Authors: Chagrov Alexander , Zakharyaschev Michael
Published: November 2023

In this chapter we return to the main question of Part IV-how to determine whether a given logic satisfies a given property-and consider it from the algorithmic point of view.
Effective algebras
Authors: Stoltenberg-Hansen V , Tucker J V
Published: November 2023

The theory of the computable functions establishes the scope and limits of computation by means of algorithms on the set N = (0,1,2,...) of natural numbers. It is based on the assumption that this set of data, equipped with some very simple operations, forms an algebraic structure that is computable in some absolute sense: the data are concrete objects and the operations on the data are computable. An objective of the theory is to analyse and classify the functions and sets that can be defined, using a variety of models of computation and formal systems, over some fundamental algebras of natural numbers.
Getting started
Authors: Clark Malcolm
Published: November 2023

Tex can do all sorts of very clever things, but it seems sensible to try to do all sorts of ordinary things first, and then migrate to the more esoteric later. Experience is a potential, if expensive, teacher; let us examine what happens when we run some text through Tex-An example input file, named EXAMPLE, is listed in Figure 2.1, and the result of running that file through Tex is shown in Figure 2.2. The text presented was prepared using an ordinary editor, and was simply typed in without any particular attention to line endings, or any sort of ‘formatting’. The ‘formatting’ will be Tex’s concern, not ours. The only concession to the ‘logical’ structure of the text has been to divide the text into paragraphs, by leaving a blank line at the end of each paragraph.
More Maths
Authors: Clark Malcolm
Published: November 2023

This chapter draws together some more of ‘IEX’s wide range of facilities for mathematics, filling in some of the features which provide the flexibility needed to support a very diverse area of technical typesetting. Some of these features help to lay the groundwork for an understanding of the way that ‘IEX really does work. Such an understanding may not be necessary for the vast majority of tasks to which we put ‘IEX, but may give us the confidence to believe that we can handle the exceptional or novel situations too.
Tables by tabs
Authors: Clark Malcolm
Published: November 2023

One reason for tackling matrices before tables was to demonstrate how straightforward it is to obtain aligned entries. There is very little difference between a matrix and a table - at least, not from the way it is laid out. From a mathematical point of view they may have rather different properties, if they do indeed contain numerical information. Of course, many tables will contain no numerical information at all. Two elements will be preserved in any discussion of Tex’s tabular facilities, the & and the \er. Before looking at Tex’s handling of tabular material, consider this quotation from Richard Beach (1986): Designing table typography is a hard problem. There are many formatting details to get right and there is only a small amount of space with which to work. The two-dimensional nature of tables requires alignment of information in both directions at the same time. It is very important to maintain control over placement because the organization of information in tables is part of the message. Juxtaposition and other spatial relationships within tables have an important impact on the way in which tables convey information.
The Restricted Burnside Problem
Authors: Vaughan-Lee Michael
Published: November 2023

The first edition of this book provided an account of the restricted Burnside problem making extensive use of Lie ring techniques to provide a uniform treatment of the field. It also included Kostrikin's theorem for groups of prime exponent. The second edition, as well as providing general updating, contains a new chapter on E.I. Zelmanov's highly acclaimed and recent solution to the Restricted Burnside Problem for arbitrary prime-power exponent. This material is currently only available in papers in Russian journals. This proof of Zelmanov's theorem given in the new edition is self contained, and (unlike Zelmanov's original proof) does not rely on the theory of Jordan algebras.