bluuebunny/crossref_metadata_2025_split huggingface dataset:
Second-Order Differential Equations In The Phase Plane
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
Very few ordinary differential equations have explicit solutions expressible in finite terms. This is not simply because ingenuity fails, but because the repertory of standard functions (polynomials, exposing, and so on) in terms of which solutions may be expressed is too limited to accommodate the variety of differential equations encountered in practice. Even if a solution can be found, the ‘formula ‘ is often too complicated to display clearly the principal features of the solution; this is particularly true of implicit solutions and of solutions which are in the form of integrals or infinite series.
Poincaré Sequences, Homoclinic Bifurcation, And Chaos
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
And its phase diagram in the (x, y)-plane. Let :E be a curve or cross-section of the plane with the property that it cuts each phase path transversely in some region of the phase diagram, that is, it is nowhere tangential to a phase path. It is called a Poincaré section of the phase diagram. Consider a point A 0 : (x0, y0) on the Poincaré section :E shown in Fig. 13.1. If we follow the phase path through A 0 in its direction of flow then it next cuts :E in the same sense at A 1 : (x 1, y1). This point is the first return, or Poincaré map, of the point A 0. We are not implying that such a point must exist, but if it does then it is called a first return. If we continue on the phase path, then the first return of A 1 is A 2 : (x2, y2) (see Fig. 13.1). We can represent this process as a mapping or function by an operator P:E which is such that for the particular
Perturbation methods
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
Trigonometric identities The following identities are useful in the application of perturbation methods. The constants A0 and B0 must be put equal to zero since otherwise x1 will include non-periodic terms. As in (i), complementary functions must be put equal to zero at each stage to eliminate nonperiodic terms. As in (i), complementary functions must be put equal to zero at each stage to eliminate nonperiodic terms. When ω0 is given, find for what values of ω there are three possible responses. (Find the stationary values of F with respect to a0, with ω fixed. These are the points where the response curves of Figure 5.4 (in NODE) turn over.) Find the equivalent linear form (NODE, Section 4.5) of the expression ¨x +_2x−εx3, with respect to the periodic form x = a cos t . Use the linear form to obtain the frequency– amplitude relation for the equation Solve the equation approximately by assuming that a = a0 +εa1, and show that this agrees with the first harmonic in NODE, eqn (5.23). (Note that there may be three solutions, but that this method of solution shows only the one close to {_/(1 − _2)} cos t .)
Forced oscillations: harmonic and subharmonic response, stability, and entrainment
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
which is the polar equation of phase paths in the van der Pol plane for large r. Since | 4_β cos θ|is bounded, the paths will be approximately circles for large r, which agrees with the solution of Problem 7.1. By examining the non-periodic solutions of the linearized equations obtained from the first part of Problem 7.8, show that the two solutions, period 2π, obtained are respectively stable and unstable. The presence of ‘sum and difference tones’ with frequencies ω1±ω2 can be detected in sound resonators having suitable nonlinear characteristics, or as an auditory illusion attributed to the nonlinear detection mechanism in the ear (McLachlan 1956). The iterative method of solution can be adapted to simpler forced oscillation problems involving a single input frequency.)
The existence of periodic solutions
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
A system has exactly one equilibrium point, n limit cycles and no other periodic solutions. Explain why an asymptotically stable limit cycle must be adjacent to unstable limit cycles, but an unstable limit cycle may have stable or unstable cycles adjacent to it. Let cn be the number of possible configurations, with respect to stability, of n nested limit cycles. By considering the path directions across each of the suggested topographic systems show that in each of the cases given there exists a limit cycle. Locate the region in which a limit cycle might exist as closely as possible. Show that in each case only one limit cycle exists:
Poincaré sequences, homoclinic bifurcation, and chaos
Authors: Jordan D W , Smith P 
Published: November 2023

Abstract
where r(0) r, θ (0) θ. The system has an equilibrium point at the origin, which is an unstable spiral. The system also has a stable limit cycle given by circle r 1. The polar equations of the paths are given by Two rings can slide on two fixed horizontal wires which lie in the same vertical plane with separation a. The two rings are connected by a spring of unstretched length l and stiffness μ. The upper ring is forced to move with displacement Φ(t) from a fixed point O as shown below or in Figure 13.41 (in NODE). The resistance on the lower ring which has mass m is assumed to be mk × speed. Let y be the relative displacement between the rings.
Manifolds
Authors: Holm Darryl D , Schmah Tanya , Stoica Cristina , Ellis David C P 
Published: November 2023

Abstract
A manifold is basically a smooth curve or surface or higher-dimensional analogue. The key property of a manifold is the existence of consistent local coordinate charts, for example spherical coordinates on a sphere, which allow us to do calculus on manifolds just as in Euclidean spaces R . Manifolds appear in many contexts in mechanics, including as configuration spaces.
Geometry on manifolds
Authors: Holm Darryl D , Schmah Tanya , Stoica Cristina , Ellis David C P 
Published: November 2023

Abstract
Differential geometry is geometry studied via advanced calculus. It applies not just in Euclidean space but on general smooth manifolds. Riemannian geometry is the oldest branch of differential geometry, and the branch that most directly generalizes Euclidean geometry: it is concerned with lengths and angles. In mechanics, Riemannian geometry is used to describe kinetic energy; and also the paths followed by particles in the absence of external forces, which are geodesics.
Pseudo-rigid bodies
Authors: Holm Darryl D , Schmah Tanya , Stoica Cristina , Ellis David C P 
Published: November 2023

Abstract
Consider a body that can stretch and shear as well as rotate, so that it can undergo arbitrary orientation-preserving linear transformations around some fixed point. The configuration of the object at a certain time is determined by an element of GL+(3) that maps a given reference configuration to the current one.
EPDiff solution behaviour
Authors: Holm Darryl D , Schmah Tanya , Stoica Cristina , Ellis David C P 
Published: November 2023

Abstract
This chapter discusses the coherent particle-like properties of the unidirectional singular solutions of the EPDiff equation (11.29). These singular solutions emerge from any smooth spatially confined initial velocity profile u(x, 0). After emerging, they dominate the evolution in interacting fully nonlinearly by exchanging momentum in elastic collisions. The mechanism for their emergence is proven to be pulse steepening due to nonlinearity. Several examples of the dynamics among singular solutions are given.
EPDiff in n dimensions
Authors: Holm Darryl D , Schmah Tanya , Stoica Cristina , Ellis David C P 
Published: November 2023

Abstract
This chapter discusses the n-dimensional generalization of the one-dimensional singular solutions of the EP equation studied in the previous chapter. Much of the one-dimensional structure persists in higher dimensions. For example, the parameters defining the singular solutions of EPDiff in n dimensions still obey canonical Hamiltonian equations. This is understood by identifying the singular solution ansatz as a cotangent-lift momentum map for the left action of the diffeomorphisms on the lower-dimensional support set of the singular solutions.
Computational anatomy: contour matching using EPDiff
Authors: Holm Darryl D , Schmah Tanya , Stoica Cristina , Ellis David C P 
Published: November 2023

Abstract
Computational anatomy (CA) must measure and analyse a range of variations in shape, or appearance, of highly deformable biological structures. The problem statement for CA was formulated long ago in a famous book by D’Arcy Thompson [Tho92]. In a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each.
Statistical Principles and Techniques in Scientific and Social Research
Authors: Krzanowski Wojtek J 
Published: November 2023

Abstract
This graduate-level text provides a survey of the logic and reasoning underpinning statistical analysis, as well as giving a broad-brush overview of the various statistical techniques that play a major role in scientific and social investigations. Arranged in rough historical order, the text starts with the ideas of probability that underpin statistical methods and progresses through the developments of the nineteenth and twentieth centuries to modern concerns and solutions. Assuming only a basic level of Mathematics and with numerous examples and illustrations, this text presents a valuable resource not only to the experienced researcher but also to the student, by complementing courses in a wide range of substantive areas and enabling the reader to rise above the details in order to see the overall structure of the subject.
Mathematics and authority: a case study in Old and New World accounting
Authors: Urton Gary 
Published: November 2023

Abstract
The title of an article published by Alan Bishop, ‘Western mathematics: the secret weapon of cultural imperialism’ (1990), must surely be one of the most provocative in the recent literature concerning the history of mathematics and the nature and status of mathematical practice. There are several surprises in this title, beginning with the adjective ‘western’. According to Platonism, the grounding philosophy that informs the thinking of most mathematicians, mathematical truths lie beyond human experience, in an abstract realm set apart from language, culture, and history.
The two cultures of mathematics in ancient Greece
Authors: Asper Markus 
Published: November 2023

Abstract
The notion of ‘Greek mathematics’ is a key concept among those who teach or learn about the Western tradition and, especially, the history of science. It seems to be the field where that which used to be referred to as ‘the Greek miracle’ is at its most miraculous. The works of, for example, Euclid or Archimedes appear to be of timeless brilliance, their assumptions, methods, and proofs, even after Hilbert, of almost eternal elegance. Therefore, for a long time, a historical approach that investigated the environment of these astonishing practices was not deemed necessary.
Algorithms and automation: the production of mathematics and textiles
Authors: Brezine Carrie 
Published: November 2023

Abstract
Mathematics is a diverse field. To the uninitiated, it is a mysterious subject, often presumed to be unintelligible and frighteningly abstract. Many people assume that mathematics is all about numbers, and it is; but it is also about shapes, sets, symmetry, networks, algorithms, and transformations. The specialized definitions and notational systems of mathematics make it almost impossible for a lay person to pick up a recent article on, say, knot theory and make any sense of it. But the mathematics of scholarly publications is not and has never been the only practice of mathematics.
Traditions and myths in the historiography of Egyptian mathematics
Authors: Imhausen Annette 
Published: November 2023

Abstract
How frequently it happens that books on the history of mathematics copy their assertions uncritically from other books, without consulting the sources! How many fairy tales circulate as ‘universally known truths’. (van der Waerden 1954, 6) 9is statement of van der Waerden’s rather surprised me. It was not that I did not agree with it wholeheartedly, since I have encountered just that situation many times myself. Rather, it was the fact that it came from one of the authors I blame as the source of some of the very same ‘universally known truths’ about Egyptian mathematics that I consider to be wrong.
Number, shape, and the nature of space: thinking through Islamic art
Authors: Bier Carol 
Published: November 2023

Abstract
Islamic art, when observed through the western lens of art history, is often seen as decorative and ornamental. This view sets up an opposition between Islamic art and general trends identified in western art since the Renaissance: the focus on representations of the human form, pictorial narrative, and spatial perspective. The treatment of pattern, which evolved and dourished from the tenth century onwards, is at once both universal within the Islamic world and su7ciently distinct from other cultural traditions that it may justifiably be characterized as Islamic.
Generative or Discriminative? Getting the Best of Both Worlds
Authors: Bishop Christopher M , Lasserre Julia 
Published: November 2023

Abstract
When labelled training data is plentiful, discriminative techniques are widely used since they give excellent generalization performance. However, although collection of data is often easy, the process of labelling it can be expensive. Consequently there is increasing interest in generative methods since these can exploit unlabelled data in addition to labelled data. Although the generalization performance of generative models can often be improved by ‘training them discriminatively’, they can then no longer make use of unlabelled data. In an attempt to gain the benefit of both generative and discriminative approaches, heuristic procedure have been proposed which interpolate between these two extremes by taking a convex combination of the generative and discriminative objective functions.
FDR and Bayesian Multiple Comparisons Rules
Authors: Müller Peter , Parmigiani Giovanni , Rice Kenneth 
Published: November 2023

Abstract
We discuss Bayesian approaches to multiple comparison problems, using a decision theoretic perspective to critically compare competing approaches. We set up decision problems that lead to the use of FDR-based rules and generalizations. Alternative definitions of the probability model and the utility function lead to different rules and problem-specific adjustments. Using a loss function that controls realized FDR we derive an optimal Bayes rule that is a variation of the Benjamini and Hochberg (1995) procedure. The cutoff is based on increments in ordered posterior probabilities instead of ordered p-values. Throughout the discussion we take a Bayesian perspective. In particular, we focus on conditional expected FDR, conditional on the data. Variations of the probability model include explicit modeling for dependence. Variations of the utility function include weighting by the extent of a true negative and accounting for the impact in the final decision.
Bayesian Probability in Quantum Mechanics
Authors: Schack Rüdiger 
Published: November 2023

Abstract
This paper reviews arguments for the claim that all probabilities arising in quantum mechanics are Bayesian degrees of belief. The starting point is the observation that there is a one-to-one map from quantum states to points on the probability simplex. Bayes’s rule remains valid in quantum theory, provided it is kept in mind that the outcomes of quantum measurements have no preassigned values. The probabilities of quantum measurement outcomes depend explicitly on a quantum prior whose general form will be discussed. Similar to the classical case, the quantum de Finetti theorem simplifies the analysis of repeated quantum trials if the prior is exchangeable. The Bayesian approach to quantum theory described here solves the problem of the ‘collapse of the wave function’. It also provides almost trivial explanations of a number of phenomena in quantum information theory, including the no-cloning theorem and quantum teleportation.
Bayesian Encompassing Specification Test Under Not Completely Known Partial Observability*
Authors: Almeida Carlos , Mouchart Michel 
Published: November 2023

Abstract
A Bayesian specification test based on the encompassing principle for the case of partial observability is proposed. A structural parametric null model is compared against a nonparametric alternative model at the level of latent variables. A same observability process is introduced in both models. The comparison is made between the posterior measures of the non-Euclidean parameter (of the alternative model) in the extended and in the alternative models. The general development is illustrated with an example where a linear combination of a latent vector is only observed.
Bayesian Multivariate Areal Wombling*
Authors: Ma Haijun , Carlin Bradley P 
Published: November 2023

Abstract
Multivariate data summarized over areal units (counties, zip codes, etc.) are common in the field of public health. Statistical methods for areal wombling (boundary analysis) must account for correlations across both diseases and locations. Utilizing recent developments in multivariate conditionally autoregressive (MCAR) distributions and spatial structural equation modeling, we suggest a variety of Bayesian hierarchical models for multivariate areal boundary analysis, including some that incorporate random neighborhood structure. We illustrate our methods using Minnesota county-level esophagus, larynx, and lung cancer data, using methods that permit identification of both composite and cancer-specific boundaries. Our results indicate primary boundaries in both the composite and cancer-specific response surface separating the miningand tourism-oriented northeast counties from the remainder of the state.
An Asymptotic Viewpoint on High-Dimensional Bayesian Testing*
Authors: Spitzner Dan J 
Published: November 2023

Abstract
The Bayesian point-null testing problem is studied asymptotically under a high-dimensional normal means model. Associated procedures are intended for application in finite dimensional settings, but their behavior is studied as the dimensionality grows arbitrarily. Thus, ‘high-dimensionality’ is formulated as an asymptotic concept, and interest is in the behavior of the procedures ‘in the limit’. The approach is to allow the prior null probability to vary with dimension and with prior dispersion parameters, and to guide its parameterization so that the posterior null probability behaves in accordance with Bayesian consistency concepts. Among issues studied are the objectivity of setting the prior null probability to one-half, the Jeffreys–Lindley paradox, and the influence of smoothness constraints. Relevance to applications in functional data analysis and goodness-of-fit testing is also discussed.
Stochastic Integration Theory
Authors: Medvegyev Péter 
Published: November 2023

Abstract
This graduate level text covers the theory of stochastic integration, an important area of Mathematics that has a wide range of applications, including financial mathematics and signal processing. Aimed at graduate students in Mathematics, Statistics, Probability, Mathematical Finance, and Economics, the book not only covers the theory of the stochastic integral in great depth but also presents the associated theory (martingales, Levy processes) and important examples (Brownian motion, Poisson process).
Introduction
Authors: Rendall Alan D 
Published: November 2023

Abstract
In the seventeenth century Isaac Newton showed that the way objects fall in everyday life and the motion of the moon and other celestial bodies can be explained in the context of a single theory. In this theory, the Newtonian theory of gravitation, there is a force of attraction between all material bodies. For most physical phenomena in which gravitation plays a role the Newtonian theory is sufficient for all practical purposes even today. There is, however, another description of gravitation which is more accurate in some situations. This is general relativity which was introduced by Albert Einstein in 1915.
Global results
Authors: Rendall Alan D 
Published: November 2023

Abstract
The Gowdy spacetimes were introduced in Section 4.4. As pointed out there the essential field equations are a system of two semilinear wave equations for functions P(t, θ), Q(t, θ) which are periodic in 0. The coefficients of the equations become singular at t = 0 and so if data are prescribed at t = to for some to &gt; 0 the maximal interval on which a solution could be defined is (0, ∞). In fact it is known that for any smooth data the corresponding solution does exist on that whole interval.
The Einstein–Vlasov system
Authors: Rendall Alan D 
Published: November 2023

Abstract
The phenomenological matter model for which the most mathematical results are known for the Einstein-matter system is collisionless matter solving the Vlasov equation. A variety of existing mathematical results on this subject are surveyed in this chapter. First some of the known results for related systems are summarized. Before coming to the Einstein-Vlasov system itself results on some related systems will be discussed. This is motivated by the hope that understanding of the properties of solutions of the Einstein-Vlasov system may be improved by interaction with the theory of these other equations.
Riemannian Holonomy Groups
Authors: Joyce Dominic D 
Published: November 2023

Abstract
Let M be a manifold, and g a Riemannian metric on M . Then there is a unique, preferred connection on TM called the Levi-Civita connection, which is torsion-free and satisfies g = 0. The curvature R( ) of the Levi-Civita connection is called the Riemann curvature, and its holonomy group Hol( ) the Riemannian holonomy group Hol( g) of g. In 1955, Marcel Berger proved that if ( M, g) is a Riemannian manifold with M simply-connected and g irreducible and nonsymmetric, then Hol( g) must be one of SO( n), U( m), SU( m), Sp( m), Sp( m) Sp(1), G 2 or Spin(7). The goal of 3.1– 3.4 is to explain what this result means and how it is proved. We start with the Levi-Civita connection, Riemann curvature, and Riemannian holonomy groups. After sections on reducible Riemannian manifolds and symmetric spaces we move onto Berger ‘s classification, describing the proof and the groups on Berger ‘s list. Sections 3.5 and 3.6 explore the relationship between the holonomy group Hol( g) and the topology of the underlying manifold M , in particular when M is compact.
Calibrated Geometry
Authors: Joyce Dominic D 
Published: November 2023

Abstract
The theory of calibrated geometry was invented by Harvey and Lawson in their seminal paper [151], which is still an excellent reference. It concerns calibrated submanifolds, a special kind of minimal submanifold of a Riemannian manifold M , which are defined using a closed form on M called a calibration. It is closely connected with the theory of Riemannian holonomy groups because Riemannian manifolds with reduced holonomy usually come equipped with one or more natural calibrations.
Associative, Coassociative And Cayley Submanifolds
Authors: Joyce Dominic D 
Published: November 2023

Abstract
In Chapter 4 we defined calibrations and calibrated submanifolds. We now apply these ideas to manifolds with exceptional holonomy. There are two types of calibrated submanifolds in 7-manifolds with holonomy G 2, called associative 3-folds and coassociative 4-folds, and one in 8-manifolds with holonomy Spin(7), called Cayley 4-folds. Coassociative 4-folds can be defined in terms of the vanishing of a closed form. This implies that they behave rather like special Lagrangian submanifolds in Chapter 8, so that moduli spaces of compact coassociative 4-folds are smooth, for instance.