bluuebunny/crossref_metadata_2025_split huggingface dataset:
The Diffusion and Wave Equations and the Equation of Laplace
Authors: Collins Peter J 
Published: November 2023

Abstract
The material in this chapter is in many ways of considerable significance. The equations under consideration arise from classical, and classic, problems in mathematical physics. They have provoked some of the finest work in classical analysis. The range of theory and techniques that have been invented to tackle the various instances that arise is extraordinary and has occupied some of the most celebrated figures in the history of mathematics. We have necessarily had to have rather limited objectives here. The aim is to find some important, if elementary, solutions to the equations which in turn will expose some of their main properties. Our main tool is indeed an elementary one which may already be familiar to the reader: the method of separation of variables. However, even with this tool, we find ourselves working with such well-known analytical objects as Bessel functions and Legendre polynomials. Later chapters discuss other aspects of the subject in the context of the calculus of variations, of the Sturm–Liouville equation, of complex analysis and of transform theory. The final section of this chapter fulfils a promise made in the last: we shall discuss existence and uniqueness of solutions and, what is so important these days when numerical analysis and the computer provide approximate solutions to otherwise intractable equations, whether a problem is ‘well-posed’, that is, whether its possible solutions are ‘continuously dependent on the boundary conditions’, and so, approximations are likely to be accurate.
Unidimensional theories
Authors: Pillay Anand 
Published: November 2023

Abstract
Zilber’s theorem (proved in section 4 of Chapter 2) played a fundamental role in proving ‘global’ results about totally categorical theories and/or ω-categorical, ω-stable theories. Similarly the information gained in Chapters 4 and 5 about the structure of locally modular minimal types (together with the strongly minimal/locally modular dichotomy theorem from section 3 of Chapter 2) can be used to help prove ‘global’ results about theories of finite U-rank. The theorems we prove in this chapter belong to ‘classical’ classification theory, that is concern the number of models in certain cardinalities of certain theories. However, the geometric information is crucial in their proof. Basically in this chapter we are concerned with unidimensional superstable theories. There are two main problems here: (i) the spectrum function for uncountable unidimensional theories, (ii) the number of countable models of countable unidimensional theories. We shall prove partial results in both cases.
Varieties
Authors: Howie John M 
Published: November 2023

Abstract
In Chapter 3 we explored some important links between rational languages and abstract algebra, the key notion being that of the syntactic monoid of a language. In this chapter we explore these links further. The central result is the Variety Theorem (7 .3.11) due to Eilenberg (1975) The deepest result (Theorem 7.4.9), due to Schutzenberger (1965) establishes a connection between aperodic monoids and so-called ‘;star-free’; languages.
Sequential systems
Authors: Smullyan Raymond M 
Published: November 2023

Abstract
We now come to our main topic. Our purpose in this and the remaining chapters is to unify various fixed point properties of a more complex sort than those of Chapter 2 that arise in indexed relational systems, such as that of recursion theory, applicative systems, such as combinatory logic, and representation systems, with their applications to mathematical logic. To this end we introduce the notion of sequential systems, which will be the basis for the next several chapters.
Some special fixed point properties of combinatory logic
Authors: Smullyan Raymond M 
Published: November 2023

Abstract
Although many fixed point theorems of combinatory logic are special cases of fixed point theorems of sequential systems, others are not — in fact they are not even statable in terms of sequential systems. Applicative systems have very special features which we will look at in this and the next two chapters. In our final chapter, we will consider some other fixed point theorems for combinatory logic that do have generalizations to sequential systems.
The classification of transitive G-spaces
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
The theorem of Chapter 5,that every G-space is expressible in a unique way as a disjoint union of orbits, reduces many questions about actions of groups to the study of transitive G-spaces. Our next aim is to understand transitive actions, and to produce a catalogue of all transitive G-spaces up to G-isomorphism for a given group G. This is not as hard as it might seem. We already have a good supply: the coset spaces cos(G: H) that we introduced in Example 4.2 were shown in Example 5.2 to be transitive. What we shall find now is that every transitive G-space is G-isomorphic to a coset space cos(G: H) for a suitable subgroup H of G. Thus coset spaces provide a sort of ‘canonical form’ for transitive G-spaces under G-isomorphism.
G-morphisms
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
We defined G-morphisms in Chapter 3: a morphism on the G-space Ω to a G-space Ω’ is a mapping 0: Ω -+ Ω’ that respects the action of elements of G in the sense that (for all weO and all geG. Here are some examples of morphisms other than G-isomorphisms.
Group actions in group theory
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
Most of the first seven chapters of this book have been devoted to an exposition of the pure theory of group actions. We turn now to applications of the theory. Later chapters are devoted to applications in combinatorics and geometry. This one is concerned with applications in group theory itself, and we begin by collecting together some elementary results that have already been established.
The group theory of Rubik’s magic cube
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
The cube with the mystifying ability to generate instant confusion provides an excellent example of groups acting on sets. You have probably seen the puzzle-indeed, we hope you own one. But in case not, here is a brief description. It is a cube divided into 27 small cubes, arranged 3 x 3 x 3 as they must be. Inside there is an ingenious arrangement which holds the small cubes together in such a way that any one of the ‘races’ of the magic cube may be rotated about its centre, (Fig. 19.1 ).
The twistor correspondence
Authors: Mason L J , Woodhouse N M J 
Published: November 2023

Abstract
In this chapter, we shall derive the Penrose-Ward transform, by which a solution to the ASDYM equation on a domain U in complex space-time is shown to determine, and be determined by, a holomorphic vector bundle on the twistor space P of U. For a general local analytic solution, the bundle can be represented by a patching matrix, a matrix-valued function F of three complex variables. The variables are coordinates on P, and F is the patching matrix of the holomorphic vector bundle relative to a two-set open cover of P. We explain how F can be constructed from the linear system, and how the solution to the ASDYM equation can be recovered from F by solving a lliemann-Hilbert problem.
Variation of 𝒲a With g
Authors: Dwork Bernard 
Published: November 2023

Abstract
The notation here is as in Chapter 2. In particular B( = ℤ[λ] in the applications) is a subring of K containing ℤ and the coefficients of g∈R^1 We again assume (2.6). We introduce elements of B which play the role of the resultant.
Fermat Surface
Authors: Dwork Bernard 
Published: November 2023

Abstract
The object of this section is to give the proof of Lemma 19.3.1. To simplify our notation we put n1 = l, n2 = 1 so here S = {1, 2, …, l-1, l + 1 and with no loss in generality we take d1 = 1 so that g and H are defined
Preliminaries
Authors: Evens Leonard 
Published: November 2023

Abstract
Let G be a group, and let k be a commutative ring. We shall usually assume that G is finite, and that k is a Dedekind domain; the most important cases are k = Z or k is a field. Let kG denote the group ring of G over k. All modules (except where noted) are to be left kG-modules. Since G is a group, the anti-automorphism kG → kG defined by g ↦ g −1 provides an isomorphism between the category of left kG-modules and the category of right kG-modules, so the sidedness of modules is not usually an issue. Often, when the base ring k is clear, we shall abbreviate kG by G in formulas.
Products in cohomology
Authors: Evens Leonard 
Published: November 2023

Abstract
H*(G, k) may be endowed with a multiplicative structure which turns it into a commutative graded ring. (Here, ‘commutative graded’ means that the product satisfies the graded commutation rule αβ = (‒1)Pq βα for α ∈ H P (G, k), β ∈ Hq(G, k).) Much of this monograph will be concerned with what is known about the structure of this ring and how it is related to the structure of G. At present, unfortunately, there is relatively little such knowledge.
Relations to cohomology of subgroups
Authors: Evens Leonard 
Published: November 2023

Abstract
Let G be a group and H a subgroup. If A is a kG-module, it is also a kH-module by restriction, and by functorality there is an induced map H*(G,A) → H*(H,A) which is usually called restriction and denoted resG→H. If X → k is a kG-projective resolution, since kG is kH-free, X → k is also a kH-projective resolution. This observation simplifies many proofs which were originally done in a very cumbersome manner using the bar resolution.
A sufficient condition for strong topological stability
Authors: Du Plessis Andrew , Wall Terry 
Published: November 2023

Abstract
In this chapter we present results showing that multi transversality to an appropriate submanifold partition of jet space implies c 0-stability. We will proveTheorem 9.1.1 Let . ℱ be a Jk 𝒦-invariant partition of an open subset Uℱ,y, of Jk(n, p) by civilized submanifolds. Let N, P be smooth manifolds of dimensions n, p respectively.
Introduction
Authors: Petrie Ted , Randall John 
Published: November 2023

Abstract
In this chapter we give some background material about the classification of 4-manifolds. We indicate how the intersection form of a simply connected 4-manifold determines its homotopy type, and discuss the natural questions which may be asked. Freedman’s classification of topological 4-manifolds is quoted (Theorem 1.5.1), and this is contrasted with what is known about the classification of differentiable 4-manifolds.
The spectral Lagrange-Galerkin method for the Navier-Stokes equations
Authors: Siili E , Ware A 
Published: November 2023

Abstract
The numerical solution of time-dependent convection-dominated diffusion equations is the central problem in fields such as meteorology, oil reservoir simulation, modelling of aerodynamic or geophysical flows, magneto-hydrodynamics and many others. Unfortunately, most standard numerical methods for these equations suffer from severe stability-related time step limitations. In recent years many successful numerical schemes have been introduced that exploit the ‘nearly hyperbolic’ nature of such equations by making use of the characteristic curves of the hyperbolic parts of the operator, or equivalently the particle trajectories generated by the convection term.
Numerical diffusion estimates as input for grid adaptation
Authors: Tattersall P , McGuirk J 
Published: November 2023

Abstract
Much current discussion centres on the assessment of the quality of grids for numerical calculation of aerodynamic flows. Geometric properties of a grid can easily be measured but the one really important property of a given grid is the quality of the solution which can be obtained on it. The quality of the solution must, in the end, be assessed by comparison with experimental data. However, because all algorithms approximate the complete equations of fluid motion in various ways (physical approximations such as turbulence models and the numerical approximations of the discretisation process), they cannot be expected to reproduce the physical flow-field perfectly. Thus, it is very important to check for numerical consistency of the solution before conclusions may be drawn on the physical modelling, and it is at this stage that grid quality is important. Checking for consistency means ensuring that the solution does not change appreciably with refinement of the grid, and that the solution does not contain numerical diffusion of sufficient magnitude to corrupt the prediction of the physical diffusion processes. The optimum route to grid independence will involve adaptation.
Generation and analysis of hybrid structured/unstructured grids
Authors: Childs P N , Shaw J A 
Published: November 2023

Abstract
The multiblock procedure (Shaw et al. 1988) has enjoyed considerable success in gridding many realistic aircraft configurations. In this approach, the grid is composed of a set of topologically hexahedral blocks, each containing a curvilinear mesh, and spatial block connectivities are described via a so-called block topology which can be determined automatically for target configurations. Within blocks, the mesh is determined as the solution of a set of elliptic PDEs whose source terms drive orthogonality at boundaries and mesh spacing in the field. C2 continuity is achieved across block interfaces.
Energy-reducing boundary conditions for the simulation of compressible viscous flows in domains with moving boundaries
Authors: Hegerl Gabriele C , Sachs Albert 
Published: November 2023

Abstract
Non-reflective boundary conditions are essential for many numerical flow simulations, especially when sound waves are present. This is the case for the numerical simulation of subsonic compressible flows in interior domains with moving boundaries, where the movement of the boundary will generate soundwaves. When such a flow is simulated numerically, usual boundary conditions at the in- and outflow boundary reflect the generated sound waves and cause the simulation to quickly break down. We have derived non-reflective boundary conditions for subsonic, viscous, compressible flows. In order to restrict the computational and formal effort, we only consider the two-dimensional case, the extension to three dimensions is straightforward.
Implicit multigrid algorithm for Euler equations on block-structured grids with discontinuous interfaces
Authors: Wang Lixia , Caughey David A 
Published: November 2023

Abstract
The challenge to develop algorithms to simulate accurately flows over realistic aerodynamic configurations has led to the exploration of block-structured grids. This approach substantially simplifies the task of grid generation for complex geometries and provides a natural framework for parallel computing. This approach also facilitates the use of different flow models in different blocks according to the physical characteristics of the flow, and the same applies for efficient grid refinement strategies.
A comparison of multidimensional upwinding for cell vertex schemes on triangular meshes
Authors: Rudgyard Michael 
Published: November 2023

Abstract
In recent years, several authors have studied ‘truly’ multidimensional upwind schemes based on cell vertex discretizations. Schemes for both quadrilateral meshes (Powell et al. 1989) and triangular meshes (Giles et al. 1990, Struijs et al 1991) have been considered.
Axiomatic presentation of propositional temporal connectives
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark 
Published: November 2023

Abstract
As with many logics, one of the crucially important properties of the sorts of temporal logics that we met semantically in the last chapter is that there are also syntactic ways of characterizing validity. We will look at decidability questions in a later chapter but here we will concentrate on the axiomatic presentation of temporal systems.
Predicate temporal logics
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark 
Published: November 2023

Abstract
The move from the propositional to the predicate version of a logic is simple in principle, though one has several options to consider. Temporal logic is a language for talking about several databases and the connections between them. Predicate logic talks about domains of individuals. Put together we have to deal with several domains of individuals and the question is what are the individuals of the combined domains? This is the source of our options. Technically, the move to predicate logic is undertaken by adding individual variables to the language and possibly individual constants, and adding atomic predicates and quantifiers ∀ and ∃.Thus predicate temporal logic is syntactically like classical predicate logic with the addition of the temporal connectives. The problem is what do the variables range over? Let us be more specific.
Basic many-dimensional systems
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark 
Published: November 2023

Abstract
Traditional temporal logic is one-dimensional in the sense that the semantic evaluation of formulae involves only one point of reference in an indexical way. The standard truth definition, for instance, characterizes the truth of a temporal formula with respect to a time structure (or flow of time) and a moment t in that structure. The moment t is thus taken to be the (temporal) context where the formula as a whole holds. The semantic account of temporal operators such as P and F involves changing the initial point and reduces the truth of a formula at t to the truth of its subformulae at other moments. Except for certain tricks associated with the IRR-rule, there is really no way to ‘get back’ to t—to ‘keep track’ of the initial point of evaluation.
Expressive completeness of Stavi connectives over general linear time
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark 
Published: November 2023

Abstract
In the preceding chapter we saw that the connectives S and U have the separation property and hence expressive completeness over any Dedekind complete flow of time. In the next section we will prove that {U, S} is not expressively complete over a general linear flow of time so that Since and Until alone cannot have the separation property over the class of all such flows.
Intuitionistic Logic
Authors: Chagrov Alexander , Zakharyaschev Michael 
Published: November 2023

Abstract
From the set-theoretic point of view intuitionistic propositional logic is a subset of the classical one: it can be defined by the calculus which is obtained fromClby discarding the law of the excluded middle (Al0). It is Brouwer’s (1907, 1908) criticism of this law that intuitionistic logic sterns from. However, the philosophical and mathematical justifications of these two logics are fundamentally different.
Algebraic Semantics
Authors: Chagrov Alexander , Zakharyaschev Michael 
Published: November 2023

Abstract
Algebraic semantics abstracts from the intended meaning of logical connectives and interprets them just as operations on an arbitrary setAof objects, some of which are regarded as distinguished. Each formula φ (p1, …,Pn)gives rise to a function φ(x1, ... , Xn)onA,and we may consider φ (p1, …,Pn)to be valid in this “interpretation” if fφ(a 1, ... i&gt;an) is a distinguished object, for every a1, ... ,an∈A. It is not hard to see that all our logics are complete with respect to this highly abstract semantics. But to use it profitably, we should know something about the constitution of algebras corresponding to modal and superintuitionistic logics.
Complexity Problems
Authors: Chagrov Alexander , Zakharyaschev Michael 
Published: November 2023

Abstract
Suppose we have managed to construct a decision algorithm for a given modal or si-logic. Then we are facing the following questions. What are the complexity parameters (say, the required time and memory) of this algorithm? Does there exist a simpler algorithm? Since it isa prioriimpossible to estimate the efficiency of the decision procedure provided by Harrop’s theorem, and the use of Büchi’s and Rabin’s theorems reduces the decision problem for a propositional logic to that for a second order theory (which is known to be very complicated), we consider here only those decision algorithms that are based on estimating the size of minimal frames separating formulas from logics.
Abstract interpretation: a semantics-based tool for program analysis
Authors: Jones Neil D , Nielson Flemming 
Published: November 2023

Abstract
Desirable mathematical background for this chapter includes basic concepts such as lattices, complete partial orders, homomorphisms, etc.the elements of domain theory, e.g. as in the chapter by Abramsky in volume 3 of this Handbook, or the books [Schmidt, 1986] or [Nielson and Nielson, 1992a]. the elements of denotational semantics, e.g. as in the chapter by Tennent in volume 3 of this Handbook, or the books [Schmidt, 1986] or [Nielson and Nielson, 1992a]. Interpretations as used in logic. There will be some use of structural operational semantics [Kahn, 1987], [Plotkin, 1981], [Nielson and Nielson, 1992a], for example deduction rules for a program’s semantics and type system. The use of category theory will be kept to a minimum but would be a useful background for the domain-related parts of section 3.
Basic: Cofinalities Of Small Reduced Products [Sh 345a]
Authors: Shelah Saharon 
Published: November 2023

Abstract
After giving the standard definition of cofinality of a partial order and reduced products (of ordinals) (in Def 1.1) we present one of the book’s main Definitions (1.2): for a set a of regular cardinals, pcf(a) is the set of the cardinals 0 which has the form: θ is the cofinality of π a/ D, D an ultrafilter on a. (Essentially a is always, after 1.4, a set of regular cardinals : ≥/a/ so we shall not always remember to state this).
Elementary feature logic and meta-logical concepts
Authors: Sandewall Erik 
Published: November 2023

Abstract
We proceed now to the introduction of the base logic, using a combination of two perspectives. One perspective is from the point of view of inhabited dynamical systems: given the definitions of Chapters 1-3, what logic can properly express scenario descriptions and obtain the intended conclusions according to the underlying semantics? The other perspective is in relation to previous research on common-SCDse reasoning about actions and time: how does the preSCDt material build on and extend what has been published before? I am going to alternate between those two perspectives, both in the preSCDt chapter and in those that follow.
Entailment methods for 𝒦-IA using DFL-1
Authors: Sandewall Erik 
Published: November 2023

Abstract
It is readily verified that SCD⊆ SCD’ does not imply
∑IA(〈O,II,scD’,0Bs 〉) ⊆ ∑IA(〈O,II,scn,0Bs 〉). Section 8.3 in the previous chapter defined the set of intended models, ΣIA, as a mapping from chronicles to model sets, using a time-constructive and simulation-like process. The entailment methods being analyzed here are functions which are restrictedly equivalent to 𝒦IA, that is, for some classes of chronicles they give the same value, but they are defined in terms of operations on classical models and model sets.
Duration constraints
Authors: Sandewall Erik 
Published: November 2023

Abstract
The chronological entailment criteria in the previous chapter required equidu rationality for correctness. This restriction is quite severe: if actions are allowed to have a duration in time other than 1, and if the result of the action can vary due to the initial conditions, one would certainly expect that the duration can vary as well. The present chapter will provide a method which removes this restriction.