bluuebunny/crossref_metadata_2025_split huggingface dataset:
Fubini’s Theorem and Tonelli’s Theorem
Authors: Priestley H A 
Published: November 2023

Abstract
This chapter presents two theorems which allow us to relate the integral on JR with the integral on ℝk for k &gt; l and to evaluate and manipulate higher dimensional integrals. These theorems are attributed to Fubini and Tonelli, whose names sound rather appropriately like those of a conjuring act. Tonelli’s magic spell proves integrability assertions, while Fubini’s interchanges the order of integration with respect to different variables. Like all conjuring tricks, Fubini’s and Tonelli’s theorems are not as mysterious as they might seem. They are a direct consequence of the way in which the integrals are set up. In Chapter 25 we introduced L(ℝk). Here we shall work throughout with k = 2, noting that extensions to higher dimensions are easy to formulate if needed.
The spaces L1, L2, and LP
Authors: Priestley H A 
Published: November 2023

Abstract
We have hitherto exploited the vector space structure of L, and dealt with various approximating sequences to integrable functions: Linc -sequences, truncations, etc. This chapter brings these ideas together in a systematic way and sets spaces of integrable functions in their rightful place in the mathematical orchestra. We can play here only a snatch of the overture to the symphony in which integration theory and functional analysis come together in concert. More advanced texts take up the refrain from where we are forced to leave it.
Integration in probability theory
Authors: Priestley H A 
Published: November 2023

Abstract
No account of integration theory would be complete without some discussion of its role in probability theory. This is a very large subject and we seek to do no more in this chapter than to build a bridge to specialized texts, such as [10], which we recommend for further reading. We address only readers with some prior familiarity with probabilistic ideas, at the level, for example, of [7]—space precludes us introducing with proper motivation the concepts of random variable, independence, and so on. Like Chapter 24 this is a spectators’ chapter, for which we set no exercises.
Structural Stability Of Discrete-Time Linear Hyperbolic Cocycles
Authors: Cong Nguyen Dinh 
Published: November 2023

Abstract
This chapter is devoted to the problem of structural stability of linear hyperbolic cocycles in the discrete-time case. In the deterministic case, there are many different kinds of structural stability and they are well investigated; for a bibliography we refer to Robbin [134] and Irwin [66] and the references therein. In some sense random dynamical systems can be viewed as dynamical systems which have a measure-preserving dynamical system as a factor, or, in other words, are extensions of such a factor. Hence, a crucial role will be played by the interrelation of measurability and dynamics. Therefore, random dynamical systems require not only more complicated tools for investigation, but also new approaches to them.
Structural Stability of Continuous-Time Linear Hyperbolic Cocycles
Authors: Cong Nguyen Dinh 
Published: November 2023

Abstract
We remark that in our continuous-time case the continuity in t of linear cocycles plays an important role. In the structural stability problem it allows us to prove constructively the structural stability of any uniformly contracting (or expanding) cocycle. This result is a random version of the deterministic Lemma 1.3.13 and is presented in Section 6.2. For the main result on the structural stability of linear hyperbolic cocycles we shall need some deep properties of Lyapunov scalar products and Lyapunov norms in the continuous-time case. These results are given in Section 6.3, where we also prove a theorem on canonical forms of (non-uniformly) contracting (or expanding) cocycles. In the last section we give an application of the results obtained.
Topological Invariants of Linear Cocycles
Authors: Cong Nguyen Dinh 
Published: November 2023

Abstract
It is well known that the stable and unstable subspaces of deterministic linear dynamical systems are invariant under topological conjugacies of linear systems. This follows from the topological characterization of those subspaces as the sets of points which go to the origin under the action of the system as the time tends to ±∞ while the trajectories of points on the center subspace remain bounded away from the origin. Note that the latter fact follows from the theory of Jordan normal forms of matrices. A natural question to ask is: Are the stable and unstable subspaces of linear cocycles invariant under topological conjugacies analogously to the deterministic case? The answer is ‘yes’, and in this chapter we prove that under topological conjugacies the stable and unstable subspaces of linear cocyclcs are invariant.
Inclusion maximal matchings
Authors: Karpinski Marek , Rytter Wojciech 
Published: November 2023

Abstract
In this chapter we consider matchings M which are maximal with respect to inclusion. This means that in a given graph G there is no matching M’ ≠. M such that M ⊂ M’. If X is a set of vertices then denote by Incident(X) the set of edges incident to any vertex in X. Denote also the set of edges incident with any endpoint of an edge in M by Incident(M); hence M ⊆ Incident(M). Observation Let M be a set of independent edges, and X be the set of endpoints of edges in M. Then the following conditions are equivalent: (a)M is an inclusion maximal matching. (b) E = Incident(M). (c) E = Incident(X).
Families of intervals
Authors: Karpinski Marek , Rytter Wojciech 
Published: November 2023

Abstract
A matching in a bipartite graph corresponds to a system of distinct representatives for a certain set family I. In this chapter we consider the case when the family I is particularly natural, it is a family of intervals on the same line, and the intervals are indexed by elements of a set X. Formally there are integers firstv, lastv such that I−{γυ:γυ=[firstυ⋯lastυ],υ∈X} The matching problem for convex graphs is equivalent to the problem of finding a maximum cardinality system of distinct representatives for a family I of intervals.
Matrix methods: basic classical theory
Authors: Boos Johann , Cass Peter 
Published: November 2023

Abstract
As we have already mentioned, in this and the following chapters we will deal mainly with matrix methods, in particular with their domains and the notion of convergence generated by them. Corresponding to the questions formulated in Section 1.3 we aim in Chapter 2 to give the most basic inclusion, comparison and consistency theorems. We will make use exclusively of classical (analytical) methods; some of these results we will re-prove in Parts II and III by functional analytic methods.
Matrix methods: structure of the domains
Authors: Boos Johann , Cass Peter 
Published: November 2023

Abstract
In the first part of this chapter (8.1-8.5) we continue the investigation of the structure of domains which we started in Chapter 2. Whereas in Chapter 2 we made use exclusively of classical methods we now apply functional analytic methods, which we discussed in Chapters 6 and 7. However, we should remark that within the scope of this book it is not possible to present all the details of the structure of domains. For more details we refer the interested reader to Wilansky’s book [254].
Fractured Fractals and Broken Dreams
Authors: David Guy , Semmes Stephen 
Published: November 2023

Abstract
Fractal patterns have emerged in many contexts, but what exactly is a pattern? How can one make precise the structures lying within objects and the relationships between them? This book proposes new notions of coherent geometric structure to provide a fresh approach to this familiar field. It develops a new concept of self-similarity called "BPI" or "big pieces of itself," which makes the field much easier for people to enter. This new framework is quite broad, however, and has the potential to lead to significant discoveries. The text covers a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics. Although fractal geometries arise in many different ways mathematically, comparing them has been difficult. This new approach combines accessibility with powerful tools for comparing fractal geometries, making it an ideal source for researchers in different areas to find both common ground and basic information.
Background Information
Authors: David Guy , Semmes Stephen 
Published: November 2023

Abstract
Let (M, d(x, y)) be a metric space, and let Ebe a nonempty subset of M. Suppose that we have a Lipschitz mapping f : E → R. Then we can extend f to a Lipschitz function on all of M, and with the same Lipschitz constant. This is well known, but let us quickly recall the proof. Suppose that f is L-Lipschitz.
Convergence of Metric Spaces
Authors: David Guy , Semmes Stephen 
Published: November 2023

Abstract
One of the basic tools for studying BPI spaces is to take limits of them, and to pass to weak tangents. This will enable us to make compactness arguments, which will sometimes permit us to derive quantitative statements from qualitative statements. In this chapter we discuss convergence of metric spaces in general, leaving BPI spaces to the next chapter. We discuss also convergence of mappings between spaces that are converging, convergence of subsets, and convergence of measures on converging spaces.
Weak Tangents
Authors: David Guy , Semmes Stephen 
Published: November 2023

Abstract
Now we want to apply the notions of convergence from Chapter 8 to BPI spaces. This will permit us to make some compactness arguments. Let(M,d(x,y)) be a metric space. For eachp ∈M and O &lt;t ≤ diamM we useMp,t to mean the pointed metric space(M,t‒ d(x, y),p). Notice thatMp,t always has diameter ≥ 1.
Spaces Looking Down On Other Spaces
Authors: David Guy , Semmes Stephen 
Published: November 2023

Abstract
Definition 11.1 (Looking down) Let M and N be two BPI spaces of the same dimension. We say that M looks downon N if there is a closed subset A of M and a Lipschitz mapping f :A → N such that f(A) has positive measure. This should be compared with BPI equivalence, or more precisely Proposition 7.1, in which we would ask thatf be bilipschitz. Now we are trying to split BPI equivalence into two pieces, so to speak.
Doubling Measures And Geometry
Authors: David Guy , Semmes Stephen 
Published: November 2023

Abstract
We shall discuss examples soon, but notice that Hausdorff measure of the correct dimension on any Ahlfors regular metric space is doubling. We shall often be interested in other doubling measures on Ahlfors regular spaces, for which we think of the doubling measure as some kind of controlled deformation of Hausdorff measure.We shall discuss examples soon, but notice that Hausdorff measure of the correct dimension on any Ahlfors regular metric space is doubling. We shall often be interested in other doubling measures on Ahlfors regular spaces, for which we think of the doubling measure as some kind of controlled deformation of Hausdorff measure.
Deformations Of Bpi Spaces
Authors: David Guy , Semmes Stephen 
Published: November 2023

Abstract
It is natural to try to make a notion of “continuous families” of BPI spaces. The idea of continuous families of geometries is very old. One can think of continuous families of Riemann surfaces, for instance. These might be realized concretely as algebraic varieties, as solution sets of some collection of polynomial equations, and one can get continuous families by varying the coefficients.
Some Sets That Are Far From Bpi
Authors: David Guy , Semmes Stephen 
Published: November 2023

Abstract
Proposition 19.1 There is a I-dimensional regular set F ⊆ R with the property that if A ⊆ F is closed, if f: A → F is Lipschitz, and A 0 = {x ∈ A: f(x) ≠ x}, then |f(A 0)| = 0.
Of course | · | refers to H measure here.
We already know from Section 9.5 that there are regular sets whose intersection with a BPI set always has zero measure, but this is a stronger example of non-BPI behavior. Notice that the set mentioned in Proposition 19.1 has the property that it cannot have a nontrivial intersection with a CPI set (Definition 6.10) of dimension l.
Other Boundary Value Problems
Authors: Quarteroni Alfio , Valli Alberto 
Published: November 2023

Abstract
The theory that was developed in the previous chapters for the Poisson equation and for symmetric elliptic equations can be extended to the case of a more general differential problem of the form where Ω is a bounded domain in Rd, d = 2, 3, with a Lipschitz boundary, ℒ is a partial differential operator, f is a given data, and u is the unknown solution. More precisely, we will consider: non-symmetric elliptic problems; the linear elasticity problem; the Stokes and Oseen problems for incompressible flows; a Stokes-like problem that arises from the analysis of compressible flows; the advection equation; and the time-harmonic Maxwell system.
Introduction to defect and microstructure analysis or the analysis of real structure
Authors: Fiala Jaroslav , Snyder Robert L 
Published: November 2023

Abstract
The first applications of X-ray diffraction served to reveal the periodic, regular structure of crystalline matter. However, it was soon discovered that the real structure of crystalline matter is not perfectly periodic. Gradually, it has been found that crystalline matter, as it really exists in nature and manufactured products, has many deviations from an ideal order (von Laue, 1960). In 1950, Hosemann introduced the concept of paracrystals (Hosemann, 1950a,b) to describe the structures of material which lie somewhere between ideal crystals and completely disordered, amorphous matter. So, in fact, all materials are paracrystalline. That is why it is so important to study the real aspects of the structure of solids. The various deviations from ideal crystalline order.
Use of pattern decomposition or simulation to study microstructure: theoretical considerations
Authors: Langford J Ian 
Published: November 2023

Abstract
Abstract
An overview of the diffraction effects arising from structural imperfections is given and line profiles due to the finite size of crystallites or domains are considered in detail. Structural ‘mistakes’ and lattice distortion (microstrain) are only discussed briefly, since these topics are covered more rigorously elsewhere. Although this chapter is mainly concerned with studies of microstructure based on the integral breadth, the theoretical treatment of size effects is equally applicable to an analysis based on Fourier series or on the simulation of diffraction line profiles. The last part of the chapter is concerned with a procedure based on the Voigt function for characterizing structural imperfections.
X-ray analysis of the inhomogeneous stress state
Authors: Kraus I , Ganev N 
Published: November 2023

Abstract
Stress measurement, especially that of residual stresses, represents one of the most widespread and technically important applications of X-ray diffraction. Traditionally much attention has been paid to X-ray stress analysis, namely for its possibility of reliable and effective checking of metal and alloy surface treatment technologies. That is why all the Czech X-ray laboratories are trying to keep pace in X-ray tensometry with leaders such as Germany, USA, UK, France, Japan and Russia, both in computer-assisted control of experiments and in finding new ways of interpretation of measured strains in interatomic distances.
A new whole-powder pattern-fitting approach
Authors: Scardi P 
Published: November 2023

Abstract
Abstract
A new algorithm for the modelling of crystallite size and microstrain (size/strain) effects on profile broadening has been introduced in a traditional Rietveld refinement routine for powder diffraction data. In spite of the simplifying assumptions at the basis of the method, a considerable improvement in quality and reliability of structural refinements has been achieved. This was particularly evident when the studied samples exhibited anisotropic size/strain effects, or when several phases with different microstructural features were present. The proposed method is specially suited to applications in materials science, where the structure of the present phases is frequently known, and the microstructure is the main object of study. In this case, the method can be used as a whole-powder pattern-fitting (WPPF), that takes advantage of the structural constraints of the refinement routine. Several applications are described in this chapter, mainly regarding phase mixtures and samples containing finely dispersed crystals. A further development regarding multi-layered systems is also presented: the refinement algorithm was adapted to properly consider the layered structure typical of thin films and coatings in order to obtain a direct evaluation of the phase content in terms of layer thicknesses.
Turbulent Flow Over Growing Waves
Authors: Belcher S E , Hunt J C R , Cohen J E 
Published: November 2023

Abstract
Growth and decay of ocean waves is strongly influenced by both the sheltering effect induced by turbulent stresses in the air flow, and by dynamical effects of the critical height. Here we show how the effect of the critical level can be analysed in the asymptotic framework used previously to investigate sheltering. The new analysis of the critical layer leads to results that are consistent with classical analyses of the critical level that use more specialised energy arguments, but only when the waves grow slowly. This study shows how the sheltering and critical layer effects can operate together, which indicates a new mechanism for growth of waves as they pass through a wave group.
Structure in the Atmospheric Surface Layer over Open Ocean Waves: Representation in Terms of Phase Averages
Authors: Hristov T , Friehe C , Miller S , Edson J , Wetzel S 
Published: November 2023

Abstract
We present a method of identifying the wave-coherent fields (wind velocity, stress, pressure fluctuations, etc.) in the surface layer flow over gravity waves. Wave-coherent fields are responsible for the exchange of momentum and kinetic energy between the turbulent air flow and under lying wave field, so their extraction from field experiment data is of primary interest. The method we developed uses the concept of an analytic signal based on the Hilbert transform. This method performs well even if the waves are substantially non-monochromatic. The representation reveals the role of the phase shifts of the wave-induced fields for momentum and energy exchange.
Benjamin Memorial Lecture: Side-Band, Benjamin-Feir and Eckhaus Instabilities
Authors: Stuart J T 
Published: November 2023

Abstract
Brooke Benjamin was responsible for a wealth of important and significant advances in our understanding of the dynamics of fluids in motion. Thus he worked on surface instabilities of fluid in a vibrating container, on the hydraulic jump, on the mechanics of bubbles, on shear flows over wavy surfaces, on the instabilities of the flow over a compliant surface (the “Dolphin” problem), on the instability of flow down an inclined plane, on “Vortex Breakdown”, on “Modern” mathematical methods in fluid mechanics, on “Transitions” in Taylor-vortex flows, and on Water Waves in all their glory. He was a brilliant experimentalist and theorist, with beautiful experiments and penetrating theories to his credit.
Flow of Air Over Sharp Crested Water Waves
Authors: Croft A J , Sajjadi S G 
Published: November 2023

Abstract
Croft and Sajjadi [1] constructed a model for the high speed flow of wind over a rigid periodic boundary with sharp crests, where the undisturbed flow was perpendicular to the crests. In that model, the flow separated at each crest creating a region of vortical flow on the leeward side of each wave. This vortical flow was modelled by a line vortex parallel to the crest and situated below it. The theory predicted the strength and position of each vortex and the resulting flow field.
An Attempt to Simulate the Effect of Wave Breaking on Momentum - and Mass-Transfer by Imposing Intermittent Near-Interface Transport Properties
Authors: Eifler Walter 
Published: November 2023

Abstract
A coupled 1-D air-water model is proposed for simulating momentum and high Schmidt number gas-transfer under the condition of periodic wave breaking. The model is driven by prescribed wind speed and by a concentration difference between water and air. Periodic time dependence is forced by imposing intermittent near surface transport properties simulating temporary interface disruption due to wave breaking. The model is calibrated for a range of wind speeds with respect to experimental wind drag coefficients. Then it is applied for high Schmidt !1umber gas transfer. Comparison of predicted and measured mass transfer rates shows excellent agreement for both laboratory and open sea measurements.
Parabolic subgroups
Authors: Geck Meinolf , Pfeiffer Golz 
Published: November 2023

Abstract
Subgroups that deserve special attention are those which share defining properties of their parent groups. In the case of a Coxeter group W with its distinguished generating set S, a subgroup generated by a subset of S is a Coxeter group in its own right. These 2n subgroups of W (where n = |S |) are the parabolic subgroups of Wand this chapter is devoted to their structure and properties-both individually as well as seen as a collection.
The braid monoid and good elements
Authors: Geck Meinolf , Pfeiffer Golz 
Published: November 2023

Abstract
In this chapter, we consider the braid group and the braid monoid associated with a Coxeter system (W, S). We introduce the braid group from a purely algebraic point of view, as a group given by a presentation which is similar to that of W, but where the quadratic relations are omitted. Our aim is to prove the basic results due to Deligne and Brieskorn-Saito about a normal form for the elements in the braid group. This will be achieved in Sections 4.1 and 4.2. The subsequent section deals with the application of these results to the problem of finding elements with certain “good” properties in the conjugacy classes of W (assuming W to be finite). The desired property is formulated in Definition 4.3.1; note that it refers in an essential way to the corresponding braid group. We determine, case by case, “good” elements of minimal length in the conjugacy classes of all finite Coxeter groups. These results will play a crucial role in our discussion of rationality properties and the computation of the values of the irreducible characters of Iwahori-Hecke algebras.
Characters of Iwahori-Hecke algebras
Authors: Geck Meinolf , Pfeiffer Golz 
Published: November 2023

Abstract
In the previous chapter we have seen that a generic Iwahori-Hecke algebra H associated with a finite Coxeter group W is isomorphic to the group algebra of W and that, moreover, we have a bijection between the irreducible characters of H and of W via a specialization process. We will now carry this relation further by studying in more detail the irreducible characters of H. Similar to the induction of representations from a subgroup of a finite group, one can also define a process of induction of representations from parabolic subalgebras of Iwahori-Hecke algebras. This will be discussed in detail in Section 9.1. The basic result is that this process of induction is compatible with specializations. This will allow us to transfer the results about induced characters for finite Coxeter groups (e.g., the results on characters of parabolic type in Chapter 6) to the case of Iwahori-Hecke algebras.