bluuebunny/crossref_metadata_2025 huggingface dataset:
Markov Random Field Extensions using State Space Models
Authors: Dethlefsen Claus
Published: November 2023

The class of state space models is very broad and comprises structural time series models, ARIMA models, cubic spline models, and, as demonstrated by Lavine (1999), also Markov random field models. The Kalman filter techniques are powerful tools for inference in such sequential models. Basic references on state space model methodology are Harvey (1989), West and Harrison (1997) and Durbin and Koopman (2001). In the past decade, there has been a certain interest in developing Markov chain Monte Carlo (MCMC) methods for the analysis of complex state state space models (see Carlin et al., 1992; Carter and Kohn, 1994; Friihwirth- Schnatter, 1994; de Jong and Shephard, 1996). Our approach is not based on MCMC but on iterated extended Kalman smoothing, which may be combined with importance sampling for exact simulation (see Durbin and Koopman, 2001). Using this method, we avoid the MCMC problems of ensuring that the Markov chain is mixing well and assessing whether the chain has converged. Writing Markov random field models as state space models, following Lavine (1999), makes it possible to use Kalman filter techniques to extend and analyze more complex Markov random field models. We show how to analyze such extensions and we also deliver an illustration, formulating a model for restoring digital images with focus on finding edges in the image. However, the new class of models also have applications within agricultural experiments (see, e.g., Besag and Higdon, 1999) and disease mapping (see, e.g., Knorr-Held and Rue, 2002).
State Space Models for Density Dependence in Population Ecology
Authors: Brooks Lara E Jamieson Stephen P
Published: November 2023

In this paper, we focus upon the detection and estimation of density dependence in two species of North American ducks on the basis of annual census counts taken over a 47-year period. The data comprise the counts themselves, plus an estimate of the associated standard errors. Classical approaches to investigating density dependence in population ecology from data of this sort tend to suffer from (often extreme) sensitivity to sampling errors. Here, we adopt a Bayesian approach and demonstrate how a state space model can be used to account for both measurement and process error within the analysis. We use (reversible jump) MCMC to both fit and discriminate between the competing models and investigate goodness-of-fit using both cross-validation and Bayesian p-values. We conclude that one of these species of duck appears to exhibit density dependence, whilst the other does not and discuss the implications that this new result has for the future management of these duck populations.
A Marginal Ergodic Theorem
Authors: Lavine Michael
Published: November 2023

In recent years, several papers including Gelfand and Sahu (1999), Hobert (2001), Liu and Wu (1999), Meng and van Dyk (1999) and van Dyk and Meng (2001) have appeared which use MCMC samplers based on improper posteriors to make valid inference about a proper target posterior. The following example is in Hobert (2001). It or a close relative has appeared in Liu and Wu (1999), Meng and van Dyk (1997, 1999) and van Dyk and Meng (2001).
Outlier Robust Estimation of a Finite Population Total
Authors: Pettit Lawrence I , Sugden Roger A
Published: November 2023

We discuss a Bayesian approach to estimating a finite population total when the mean of the observed sample values is assumed to be proportional to a known covariate. Observations have one of two possible variances, a standard form or an inflated form. Observations with the larger variance are termed outliers. We learn about the proportion of outliers and parameters of the model from the sample. Inference about the mean and variance of the population total is made using the predictive distribution. The frequentist properties of the procedure are analyzed for a particular finite population by examining the bias of the estimator and coverage of the confidence interval. The performance of the estimator is shown to be superior to the Horvitz-Thompson estimator and a simple Bayes estimator which allows only the standard variance.
A Method for Sequential Optimization in Bayesian Analysis
Authors: Virto Miguel , Martín Jacinto , Insua David Ríos , Díaz Arminda Moreno
Published: November 2023

Bayesian analysis frequently leads to sequential optimization problems under uncertainty. Some examples include optimal sequential design, the analysis of influence diagrams and decision trees and many Operations Research related problems in areas like maintenance or reliability. We introduce here a method to deal with such problems, within a stochastic dynamic programming context. The basic approach is a Metropolis augmented probability simulation algorithm. Once introduced, we describe how to exploit the dynamic structure of the problem in a Bellman-like approach and describe our basic algorithm. Implementation details are discussed together with an illustrative example in preventive maintenance. An appendix includes theoretical results.
Fundamentals of the finite element method
Authors: Elishakoff Isaac , Ren Yongjian
Published: November 2023

Everybody nowadays knows what finite elements are: they are the methods for solving boundary value problems in which one divides the domain of the problem into little pieces over which the solution is approximated using polynomials. The little pieces are finite elements, and the polynomials are called shape functions—J. T. Oden (1990). … I wanted to have a better name for the method because ‘direct stiffness ‘ did not seem to convey the essence of the idea. That name is suitable to describe the procedure for assembly of discrete elements of any type, but it does not imply anything about stress analysis. So thinking of the work we had done at Boeing in comparing discrete element assembly results with finite different results, the name ‘finite element method ‘ seemed to be appropriate … it seems to have been a good choice because it has stood the test of time—R. W. Clough (1994).
Epilogue
Authors: Elishakoff Isaac , Ren Yongjian
Published: November 2023

‘It is this author ‘s belief that with the large scale introduction of computers in the practice and lately also at many technical schools, the teaching of and the approach to solving technical problems has been shifting in the wrong direction … the insight of how structures behave under loading of the older generation is being more and more replaced by the nearly religious faith of the younger ones in the predictions of their favorite computer codes. ‘
The big-bang models
Authors: Rowan-Robinson Michael
Published: November 2023

To describe the properties of a gas we do not need to study the behaviour of individual atoms and molecules. Instead we define various macroscopic quantities-density, pressure, temperature-and study the relations between these. In the same way, we make no attempt to incorporate individual galaxies, or clusters of galaxies, into our description of the universe as a whole. We imagine the matter in the universe smeared out into an idealized, smooth fluid, which we call the substratum .
Observational cosmology
Authors: Rowan-Robinson Michael
Published: November 2023

The isotropic 2.7 K blackbody radiation, the 24 per cent cosmic helium abundance, and the similarity between the ages of galaxies and the age of the universe all support the big-bang models derived from general relativity assuming the cosmological principle. Can we test these models in more detail? In particular, can we determine the current value of the deceleration parameter q (Section 4.7) and thereby deduce what the future of the universe will be? In this chapter we look at a variety of cosmological tests, using discrete sources of radiation and the microwave background radiation fluctuations, that have been applied to try to answer this question.
Algebraic sets and algebraic groups
Authors: Geck Meinolf
Published: November 2023

In this first chapter, we introduce the basic objects: algebraic sets in affine space, the corresponding affine algebras, and algebraic groups. We proceed in a completely elementary way and assume only some very basic knowledge about groups, polynomial rings, quotient rings, and fields. The general theory will be mainly illustrated with examples concerning hypersurfaces and algebraic groups.
Algebraic representations and Borel subgroups
Authors: Geck Meinolf
Published: November 2023

One of the principal aims of this chapter is to prove the conjugacy (and further basic properties) of Borel subgroups, that is, maximal closed connected solvable subgroups in an affine algebraic group.
The Neural Bases of Subliminal Priming
Authors: Dehaene Stanislas
Published: November 2023

Psychologists have long reported that words that are made invisible by forward and backward masking can nevertheless cause behavioral priming effects. Functional neuroimaging can now be used to explore the neural bases of masked priming. Subliminal priming causes reduced activation in multiple areas (fusiform gyrus, intraparietal sulcus, and motor cortex), in direct correspondence with behavioral manifestations of priming at the orthographic, semantic, and motor level. This implies that a whole stream of processors can operate unconsciously. The neural code in each area can be assessed by varying prime-target relations. A simple mathematical framework is proposed that tentatively relates priming at the voxel level with the shape of the tuning curves of single neurons in the underlying tissue.
The Influence of Scene Organization on Attention: Psychophysics and Electrophysiology
Authors: Valdes-Sosa Mitchell , Bobes Maria A , Rodriguez Valia , Acosta Yanelis , Pérez Alejandro , Iglesias Jorge , Borrego Mayelin
Published: November 2023

Many studies have examined the neural mechanisms of visual attention by recording event-related potentials (ERPs). In most of these studies, objects are abruptly presented to the observer, often in a rapid succession of events. Similar stimuli are used in the psychophysical technique known as rapid serial visual presentation (RSVP). We argue that in RSVP (and related ERP paradigms) it is difficult to evince object-based attentional effects and to study the influence of scene organization. Rapid serial object transformation (RSOT) is offered as an alternative.
Introduction
Authors: Gibson David J
Published: November 2023

Each grass-covered hillside is an open book for those who care to read. Upon its pages are written the conditions of the present, the events of the past, and a forecast of those of the future. Some see without understanding; but let us look closely and understandingly, and act wisely, and in time bring our methods of land use and conservation activities into close harmony with the dictates of Nature. The purpose of this first chapter is to introduce the grassland biome. Grasslands are the most extensive, arguably the most useful to human society, yet the most threatened biome on the planet. Nevertheless, it is surprisingly difficult to unambiguously define grassland. Here we seek such a definition (§1.1); discuss where grasslands occur (§1.2); how they are being lost, fragmented, and degraded (§1.3); and summarize their outstanding and immense value (§1.4). The chapter finishes with a brief biography of John Bews and John Weaver, two of the early pioneers of grassland ecology (§1.5).
Linear programming
Authors: Webster Roger
Published: November 2023

This chapter gives a full introduction to both the theory and practice of linear programming and game theory. The important role played by ideas from convexity is Pmphasized.
Order, Lattices, And Boolean Algebras
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

In the current chapter, we present a very special and useful class of mathematical structures, which play an important role in the algebraic description of logic. These structures, called lattices, can be characterized both as relational structures (pertaining to the notion of implication), and as algebraic structures (pertaining to the notions of conjunction and disjunction)..
Syntax
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

It is customary to think of sentences concretely as utterances stretched out linearly in time, or, even more commonly, as inscriptions stretched out linearly in space, but this very sentence is a counter-example to such over-simplicity (because of the need for line breaks).
Semantics
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

We have now discussed the mathematical theory of propositions, which we formulated in the framework of lattice theory, and we have discussed the mathematical theory of sentences (formal syntax), which we formulated in the theory of universally free algebras. In the present chapter, we discuss the relation between sentences and propositions, which is the central focus of algebraic formal semantics.
Classical Propositional Logic
Authors: Dunn J Michael , Hardegree Gary M
Published: November 2023

Classical propositional logic has associated with it a number of different algebraic log ics, which we will call Frege logic, Boolean logic, and unital Boolean logic. Although these logics are characterized by different classes of logical matrices, as we will see later, they are all strongly equivalent, which is to say that they validate exactly the same asymmetric consequences. However, they do not validate all the same symmetric con sequences.
Codes and Curves
Authors: Hirschfeld J W P
Published: November 2023

In 1981 Gappa [5] gave a construction of linear codes beginning from an algebraic curve over the finite field GF(q).The parameters of these codes are restricted by the Riemann-Roch theorem. They also give asymptotic codes better than those previously known. Here we describe the construction and give some of the subsequent results for curves of genus zero and one. Foe othe, smvey,, see [2], [7], [10], [16], [29], [30], [32], [35].
The Homogeneous Linear Equation and Wronskians
Authors: Collins Peter J
Published: November 2023

The main aim of this chapter will be to use Theorem 4 of Chapter 2 – and specifically both existence and uniqueness of solutions – to develop a theory that will describe the solutions of the homogeneous linear second-order equation where p  0; p  1; p  2 are continuous real-valued functions on [a, b] and p  2(x) < 0 for each x in [a, b]. (‘Homogeneous’ here reflects the zero on the right-hand side of the equation which allows λy to be a solution (for any real constant λ) whenever y is a given solution.) The language of elementary linear algebra will be used and the theory of simultaneous linear equations will be presumed. Central to our discussion will be the Wronskian, or Wronskian determinant: if y  1 : [a, b] → ℝ and y  2 : [a, b] → ℝ are differentiable functions on the closed interval [a, b], the Wronskian of y  1 and y  2, W(y  1; y  2) : [a, b] → ℝ, is defined, for x ∈ [a, b], by If y  1 and y  2 are solutions of (1), it will turn out that either W(y  1, y  2) is identically zero or never zero in [a, b].
The Non-Homogeneous Linear Equation
Authors: Collins Peter J
Published: November 2023

In this chapter we shall consider the non-homogeneous second-order linear equation where f : [a, b] → ℝ and each p  i: [a, b] → ℝ are continuous, and p  2(x) > 0, for each x in [a, b]. Any particular solution of (1) (or, indeed, of any differential equation) is called a particular integral of the equation, whereas the general solution c  1  y  1 + c  2  y  2 of the corresponding homogeneous equation given by Proposition 6 of Chapter 3 (where y  1, y  2 are linearly independent solutions of (2) and c  1, c  2 are arbitrary real constants) is called the complementary function of (1). If y  P is any particular integral and y  C denotes the complementary function of (1), y  C + y  P is called the general solution of (1). We justify this terminology by the following proposition, which shows that, once two linearly independent solutions of (2) are found, all that remains to be done in solving (1) is to find one particular integral.
The Diffusion and Wave Equations and the Equation of Laplace
Authors: Collins Peter J
Published: November 2023

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.
Introduction
Authors: Pillay Anand
Published: November 2023

Geometric stability theory is a subject that has grown in an organic fashion out of categoricity theory and classification theory. I will try to describe, in the next few paragraphs, and with a minimum of technical language, both the subject and its development, insofar as it is treated in the main body of the book.
Unidimensional theories
Authors: Pillay Anand
Published: November 2023

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.
Regular types
Authors: Pillay Anand
Published: November 2023

The purpose of this chapter is to generalize much of the preceding results on minimal or U-rank 1 types to regular types (in a stable theory). This includes the classification of the geometries attached to locally modular regular types. In the case of a minimal type p, dimension or U-rank was the crucial tool.
The syntactic monoid
Authors: Howie John M
Published: November 2023

In this chapter we obtain another characterization of rational languages in terms of a monoid Syn(L), called the syntactic monoid of L, associated with every L ⊆ A*. The computation of Syn(L) is then discussed, via a notion important in its own right, namely the minimal automaton of L.
Varieties
Authors: Howie John M
Published: November 2023

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

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.
Multiple fixed point properties
Authors: Smullyan Raymond M
Published: November 2023

In this chapter we unify various “double” fixed point properties, and other multiple fixed point properties that arise in recursion theory, combinatory logic, and metamathematics.
Some special fixed point properties of combinatory logic
Authors: Smullyan Raymond M
Published: November 2023

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

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

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

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.
Affine geometry
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

In the previous chapter we laid the foundation for the following formal definition:

An affine geometry 𝒜(F) of n dimensions over a field Fis a G-space (Ω, G) equivalent to (Fn, AGL(n, F)).
The group theory of Rubik’s magic cube
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C
Published: November 2023

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 ).
Symmetric Functions
Authors: Macdonald I G
Published: November 2023

Many of the objects we shall consider in this book will turn out to be parametrized by partitions. The purpose of this section is to lay down some notation and terminology which will be used throughout, and to collect together some elementary results on orderings of partitions which will be used later.
The twistor correspondence
Authors: Mason L J , Woodhouse N M J
Published: November 2023

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.
Algebraic Theory
Authors: Dwork Bernard
Published: November 2023

We define R as in Chapter 1 and H, H  0, H  ℚ in the same way. Again let
Variation of 𝒲a With g
Authors: Dwork Bernard
Published: November 2023

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

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

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.
Explicit resolutions
Authors: Evens Leonard
Published: November 2023

It is sometimes possible to calculate cohomology efficiently by a clever choice of resolution. This observation was originally motivated by the fact that calculating the cohomology of a space is often facilitated by choosing an appropriate cellular decomposition. (See Brown (1982, Chapter 1).) However, it often possible to approach the issue purely algebraically.
Products in cohomology
Authors: Evens Leonard
Published: November 2023

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

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.
General introduction
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

In this book we examine the topology and geometry of topologically stable smooth mappings. Amongst other results we describe some necessary and some sufficient conditions for topological stability. These conditions are shown to match up in the ‘seminice dimensions’ (except for one dimension-pair (19, 20)) to characterize topological stability there.
A sufficient condition for strong topological stability
Authors: Du Plessis Andrew , Wall Terry
Published: November 2023

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

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 Virtual Moduli Space β Of Virtual Connections
Authors: Petrie Ted , Randall John
Published: November 2023

We shall use the following notation. Let M be a smooth closed oriented 4-manifold with H1M; Z2) = 0 and positive definite intersection form, and E an SO(3)-vector bundle over M.
The Dirichlet Divisor Problem
Authors: Heath-Brown D R
Published: November 2023

Clearly f is non-negative, and since it extends to an entire function on the complex plane its zeros form a discrete set.
Free Lie Algebras
Authors: Reutenauer Christophe
Published: November 2023

Although Lie polynomials first appeared at the turn of the century, there have been many recent developments especially from the point of view of representation theory. This book covers all aspects, with emphasis on the algebraic and combinatorial point of view as well as representation theory.
The spectral Lagrange-Galerkin method for the Navier-Stokes equations
Authors: Siili E , Ware A
Published: November 2023

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

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

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

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

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

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

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

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

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

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

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

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>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

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

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

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

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

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

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.
Entailment methods for 𝒦-IA using occlusion
Authors: Sandewall Erik
Published: November 2023

Compared to the list of requirements for the correct applicability of PCM, we have now removed the requirements of initial-time observations and equidurationality, but the compatibility requirements remain. Reformulate r.Construct a new entailment method. Change the base logic. In previous chapters we have used the second of these options and, to a certain extent, the first one. The third option would represent more force than is necessary. For the remaining restrictions it may be possible to use reformulation in some cases, but for full generality it seems necessary to modify the base logic. The present chapter considers these alternatives.
Composite actions
Authors: Sandewall Erik
Published: November 2023

Chapter 6 defined an occurrence language which only allows elementary oc currence expressions, so that each action statement refers to a single action with its timing information and choice of arguments. An eventual general ization to the case of composite action statements was mentioned, and it is the topic of the present chapter. We shall define the language for composite actions, and extend the entailment methods and their assessments to the broader range of action statements.
Dynamic light scattering from dense polymer systems
Authors: Fytas George , Meier Gerhard
Published: November 2023

Photon correlation spectroscopy (PCS) is a powerful light scattering technique, widely used mainly because of its broad time range, 103-10-7 s, which is of great interest in studying polymer dynamics, and its versatility. In particular, PCS is extensively utilized to investigate transport properties in polymer solutions, for which good-quality samples can readily be prepared and hence the beating conditions on the detector are well-defined. PCS is much less used to study molecular motions in concentrated and undiluted polymers in the rubber and glassy state, despite their technological relevance. The requirement for good samples with intrinsic light scattering, the presence of broad distributions of relaxation times, and the lack of analytical form for the desired field autocorrelation function are probably the main reasons for the restricted number of applicationsJl-31 In this chapter, we review recent PCS applications to the study of dynamics in dense polymeric and molecular systems. In Section we present the basic physical principles of dynamic light scattering by density, concentration and optical anisotropy fluctuations in the medium at thermal equilibrium. The association of these fluctuations with localized and diffusional processes in various systems and states will be discussed in Sections 9.4 and 9.5. The analysis of time correlation functions is dealt with in Section 9.3, and slow solvent orientation in polymer solutions in Section 9.6. Other aspects of polymer solutions, extensively discussed elsewhere in this volume, will not be included.
Mathematical Representations Of Motion And Force
Authors: Meli Domenico Bertoloni
Published: November 2023

In the previous chapter we began to appreciate the links between mathematics and mechanics. One of the issues I have mentioned concerns the dimensions of physical quantities in relation to Leibnizian differentiation and integration, or to Newton’s fluxions and fluents. The connections between pure mathematics and mechanics—significantly a branch of the mixed or applied mathematics—are certainly broader and deeper than this example may suggest, and are central to my understanding of Leibniz’s theory and war with Newton. Already in Galileo with the notion of ‘momento’, or in Hobbes with the notion of ‘conatus’, the infinitely small was interwoven with the study of nature. By the end of the seventeenth century the analysis of curvilinear motion was gaining a prominent position in mechanics, and not surprisingly it interacted with the mathematical notion of curve.
Publication
Authors: Meli Domenico Bertoloni
Published: November 2023

The Tentamen appeared in the Acta Eruditorum of February 1689. At the same time Leibniz left Vienna for Venice, where he arrived on 4 March. Therefore, the essay was certainly composed in the Imperial capital. We have the manuscript draft, the final manuscript version having probably been sent to the editors of the Acta in Leipzig. The text would have been sufficiently challenging even without several misprints. Moreover, the figure was lost and had to be redrawn by Christoph Pfautz, the mathematician collaborating with the editor of the Acta Otto Mencke. As a result, the segment 3MG and the letters G at the bottom of the figure were omitted. On the basis of a manuscript diagram, I have restored the figure inserting the missing letters in angle brackets. In the footnotes I indicate Leibniz’s later emendations and explain some technical points.