bluuebunny/crossref_metadata_2025_split huggingface dataset:
Topological spaces
Authors: Gilbert N D , Porter T 
Published: November 2023

Abstract
The purpose of this chapter is to provide the theoretical background for a rigorous discussion of knots and surfaces, against which the vaguely expressed ideas lying behind our previous discussion can be made precise. The central notion is that of a topological space. The precision that topological spaces bring to the ideas of continuity and nearness is, of course, essential to any proper treatment of knots and surfaces. Moreover, topological ideas will prove crucial in the development of more sophisticated invariants in later chapters. Since our interest in topological spaces stems from specific applications that we already have in mind, we make no attempt to give a comprehensive account of even the basic theory of topological spaces here. There are several textbooks that do the job well, and we mention Armstrong (1983) and Brown (1988) as providing suitable further reading.
Presentations of groups
Authors: Gilbert N D , Porter T 
Published: November 2023

Abstract
We need group theory so as to be able to handle some of the deeper invariants of knots, in particular the group of a knot. We might expect the type of group theory needed would be fairly limited and somewhat specialized. However, the overall process translating geometric problems into algebraic ones is very much two way-algebra pervades the topology and vice versa-so quite a lot of group theory turns out to have implications in topology, whilst the topology can provide new methods and insights in group theory. Attempts to compartmentalize mathematics fail; there are too many important links joining apparently distinct branches. There is an underlying and all-pervading logical structure that is difficult to pin down and to see clearly, but which is none the less there behind all of mathematics.
Van Kampen’s theorem
Authors: Gilbert N D , Porter T 
Published: November 2023

Abstract
In this chapter we will examine the statement and proof of the Van Kampen theorem. If you glance forward to where this theorem is stated in full and in detail, you may wonder why such an apparently technical theorem is needed. Complicated theorems occur at numerous places within mathematics courses at this level and students often ask themselves why such theorems are deemed necessary. Again it is often the case that the lecturers have used the results explicitly or implicitly so many times in research projects that it becomes selfevident to them that these theorems are important and hence it can be extremely difficult to explain their importance to someone meeting them for the first time. With the Van Kampen theorem, we are lucky. We have been developing the theory of the fundamental group and if we pause to take stock of our present knowledge (and ignorance), it should become clearer why theorems of the Van Kampen type are needed. An even closer look will suggest what it should say and will also give us a sketch proof. Writing out a formal proof is another matter and will be dealt with later on; the first priorities are to guess what the result will say (before we meet its detailed statement), to see why the result is true and why certain technical restrictions are necessary; the details of the formal proof are (initially) somewhat less important. Once you understand the idea of the proof the details can be seen merely as the ‘justification of the obvious’.
Rank-1 rules-the method of good lattice points
Authors: Sloan I H , Joe S 
Published: November 2023

Abstract
In this chapter we look at rank-1 rules, which are rules of the form where z is an s-dimensional integer vector having no factor in common with N. Such rules were first introduced by Korobov (1959), Bahvalov (1959), and Hlawka (1962). They are often referred to as number-theoretic rules since their accuracy is related to the number-theoretic properties of N and the components of z (for example, see Korobov 1963 or Hua and Wang 1981).
Surface properties: electronic
Authors: Prutton M 
Published: November 2023

Abstract
The earlier chapters have concentrated upon the techniques required to obtain information about what kind of atoms are present at a surface and where they are situated. As the need for other techniques arises they will be discussed, but now attention will be turned towards the ways in which the properties of surfaces can be related to their composition and structure. The calculation of the electronic states at a surface is bound to be more complex than the corresponding calculation for electrons inside a solid. The latter is complicated enough, because the least tightly bound electrons are free to move in the periodic potential established by the arrangement of nuclei and tightly bound electrons the ion cores. Furthermore, the moving electrons repel each other because of their negative charges. Thus, Schrodinger’s equation has to be solved for a many-body periodic potential. This is rarely analytically possible. Various approximations are sought whose complexity depends upon the physical property being described and the precision needed to compare this property with experimental measurements. Free electron theory treats the periodic arrangement of positively charged ion cores in a crystal as a smeared out, generally attractive, potential for the gas of free electrons.
Surface properties: atomic motion
Authors: Prutton M 
Published: November 2023

Abstract
Up to this point the discussion of both methods and properties has been in terms of rigid lattices of atoms or molecules. In practice, of course, the atoms are in motion, and this motion should be included in a treatment of any properties it may affect. In this chapter the atomic vibrations at a surface will be discussed, as they reveal themselves in the temperature dependence of the intensity of a LEED pattern and in special features in electron-loss spectra. More extreme atomic motions occurring during the melting of a surface and during the motion of an adatom across it will also be outlined. It is well known that there is a fall in the intensity of the beams in an X-ray diffraction experiment (using a bulk single crystal) as the crystal temperature is raised. At the same time the intensity in the diffuse background of the diffraction pattern becomes higher. The simplest explanation of these observations is that the individual atoms of the crystal are vibrating independently about their equilibrium positions and, as a result, the exact Bragg conditions is not met. This is because scattered waves from the rigid lattice that were adding up in phase now have phase differences fluctuating with time due to the motion of the scatterers.
The Gauss Linking Number in Quantum Gravity
Authors: Gambini Rodolfo , Pullin Jorge 
Published: November 2023

Abstract
We show that the exponential of the Gauss (self-)linking number of a knot is a solution of the Wheeler-DeWitt equation in loop space with a cosmological constant. Using this fact, it is straightforward to prove that the second coefficient of the Jones polynomial is a solution of the Wheeler-DeWitt equation in loop space with no cosmological constant. We perform calculations from scratch, starting from the connection representation and give details of the proof. Implications for the possibility of generation of other solutions are also discussed.
Oxford Studies In Probability 4
Authors: Petrov Valentin V 
Published: November 2023

Abstract
This book is devoted to limit theorems and probability inequalities for sums of independent random variables. It includes limit theorems on convergence to infinitely divisible distributions, the central limit theorem with rates of convergence, the weak and strong law of large numbers, the law of the iterated logarithm, and also many inequalities for sums of an arbitrary number of random variables. The aim of the book is two-fold. First, the book can be a basis for a course in probability for graduate or advanced undergraduate students. Second, the book can serve as a reference formulations of many (mostly recent) results which are usually inaccessible in book form. These supplements border on the basic text and make it possible for the reader to look at the frontier of modern investigations.
Weak limit theorems: the central limit theorem and the weak law of large numbers
Authors: Petrov Valentin V 
Published: November 2023

Abstract
In Chapter 3 we studied the conditions under which the distributions of sums of independent random variables will converge to a limit distribution. We were interested in the weak convergence of distributions; the corresponding limit theorems are called the weak limit theorems, contrary to the strong limit theorems making use of the convergence almost surely. The necessary and sufficient conditions were found for the weak convergence of distributions of sums of independent random variables to a given infinitely divisible distribution. These conditions are quite complicated but are very helpful when we are looking for simpler conditions of convergence to a prescribed infinitely divisible distribution such as a normal, a degenerate, or a Poisson distribution.
The Transmission Problem Revisited
Authors: Dassios George , Kleinman Ralph 
Published: November 2023

Abstract
In this chapter we present another derivation of the low frequency expansions for transmission problems. In contrast to the problems involving impenetrable scatters, the existence of fields in v- in transmission problems enables an alternate derivation of low frequency expansions employing volume integrals over v-. In many respects the procedure for determining the coefficients in low frequency expansions is simpler than that based entirely on boundary integrals. In acoustics the lossless case was treated in Dassios (1977) and the lossy case was considered in Dassios (1989b) but not in the explicit form presented here. The development in acoustics and electromagnetics has not appeared in the literature, while the material on elasticity is based on the work of Kiriaki and Polyzos (1988).
Rayleigh Scattering
Authors: Dassios George , Kleinman Ralph 
Published: November 2023

Abstract
In this chapter we particularize the general results of Chapters 3 and 4 to the first nontrivial terms in the low frequency expansions known as Rayleigh approximations or Rayleigh terms of both the near and the far fields. Also included are the leading terms in the scattering cross sections. These results are given for general shapes in terms of the potential functions to which they have been reduced in Chapters 3 and 4. For acoustics and electromagnetics, most of the material in this chapter refers to results obtained in the 1960s and the 1970s by a number of researchers such as D. S. Jones, R. E. Kleinman, T. B. A. Senior, V. Twersky, J. Van Bladel and many others. The corresponding material in elasticity was developed in the 1980s by G. Dassios, K. Kiriaki, D. Polyzos and others.
The Ellipsoid
Authors: Dassios George , Kleinman Ralph 
Published: November 2023

Abstract
In this chapter we present explicit results for low frequency scattering by a triaxial ellipsoid with semi-axes a 1 &gt; a 2 &gt; a 3 &gt; 0. Results for rotationally symmetric ellipsoids are easily obtained by setting a 2 = a 3 (prolate spheroid) or a 1 = a 2 (oblate spheroid). Results for the sphere may be recovered by a limiting process from either a prolate or an oblate setting, but this is not always a trivial calculation due to the occurrence of indeterminate forms. No such difficulties occur in the reduction of ellipsoids to spheroids.
Spin Precession and Spin Depolarization
Authors: Karlsson Erik b. 
Published: November 2023

Abstract
The major part of this book will deal with techniques and experimental results from DPAC and μSR, which are methods based on the observation of the spin precession and spin relaxation of nuclei and positive muons respectively. This chapter includes general equations of spin motion common to NMR and EPR spectroscopies as well as to the methods of interest here, some useful analogies between NMR, DPACt and μSR, and a discussion of the number of probe spins necessary to record useful spectra with each of these methods. Since both DPAC and μSR signals are detected by radiation emitted from probe nuclei or particles which themselves are radioactive, there will also be a discussion of how these radiations are detected most efficiently with a view to extracting maximum information on the solid state systems probed.
Defects in Metals
Authors: Karlsson Erik b. 
Published: November 2023

Abstract
Isolated impurity atoms in an otherwise perfect metallic crystal form point defects in the crystals. Heavy atoms enter the host metal lattices substitutionally, and the point symmetry at such sites is the same as that for the host atoms. The EFGs at the nuclei of impurity atoms depend on the local electronic structure around these centres as well as on the symmetry and distances to the charged ion cores in the host crystal. Therefore data on EFGs contain implicit information on the electronic structure of the point defects. EFGs at point defects cannot be straightforwardly predicted from the parameters determined in a usual band-structure calculation, since such a calculation would require a detailed knowledge of the wave functions in the vicinity of the impurity atom. A theoretical approach based on a simulation of the ‘impurity plus nearest host metal atoms’ by a sufficiently large molecular cluster is a more appropriate basis for predictions of impurity EFGs and has been attempted in a few cases (Lindgren 1986). However, most of the comparison between experimental data and theory is still at a much less ambitious level. Since the experimental data are of fairly high accuracy, they can be said to form a future resource for testing electronic structure calculations for impurities in metals-a topic of considerable interest for materials science.
Semiconductors
Authors: Karlsson Erik b. 
Published: November 2023

Abstract
The physics of semiconductors is a vast field, rich in physical phenomena and extremely fertile when it comes to technical applications. The methods discussed here represent an almost vanishing fraction of the total number of works on semiconductors; however, they can be said to observe some of the phenomena from a new and different point of view and therefore provide interesting complementary information. PAC and μ.SR spectroscopies have both contributed: the first mainly as a tool to study the existence and geometry of certain defect centres under specified annealing conditions, and the second mainly as a way to mimic H impurities in Si, Ge, and 111-V (GaAs-type), II-VI (CdTe-type), and I-VII (CuCl-type) semiconductors. Both methods have also been used to observe changes in the properties of such defect centres under different p- or n-doping conditions.
Hydrogen and Hydrogen-like Impurities in Metals
Authors: Karlsson Erik b. 
Published: November 2023

Abstract
In Chapter 8 the properties of hydrogen-like impurities and their interaction with heavier trace elements in Si were discussed in some detail. The physics of hydrogen impurities in metals is essentially different: they do not form paramagnetic centres, they are much more mobile (and are sometimes described as a lattice gas filling the metallic lattice at higher concentrations), and as a consequence of the mobility they are easily trapped by other fixed defects in a metallic solid.
Quantum Diffusion of Light Interstitial Particles
Authors: Karlsson Erik b. 
Published: November 2023

Abstract
In the introduction to Chapter 9, as well as in the examples discussed earlier, we have referred to a set of potential wells for hydrogen and positive muons provided by the different interstitial sites. Between these wells there are potential barriers which the particles must overcome during diffusion. The activation energies quoted in the examples are a rough measure of the heights of these barriers.
The Quantum Theory Of Potential Scattering
Authors: Fröbrich P , Lipperheide R 
Published: November 2023

Abstract
Potential scattering is the scattering of two collision partners whose mutual interaction is described by a static potential. This potential depends in general on the structure of the collision partners, in particular, on their size and shape (usually taken as spherical), but once the potential is specified and the centre-of-mass motion is separated out, the scattering problem in the remaining relative coordinate reduces to the description of the motion of a point-particle with a mass equal to the reduced mass of the two collision partners in the static field of the interaction potential. Only elastic scattering can be described in this way.
Semiclassical Scattering
Authors: Fröbrich P , Lipperheide R 
Published: November 2023

Abstract
Although the nuclear scattering systems are so small that their quantitative description must generally be based on quantum theory, it is often found that a classical treatment is quite adequate. In fact, it was the application of purely classical laws of motion to the scattering of a-partidcs which led Rutherford to the discovery of the nucleus in 1911, years before the corresponding quantum laws had been formulated. Therefore, the question arises whether quantum scattering theory can be formulated in such a way that the connection with classical mechanics is uncovered; and whether one may then define conditions under which the quantum expressions can be approximately written in terms of purely classical quantities. This is indeed the case. The corresponding approximation is the semiclassical approximation. The condition under which it is valid is essentially that the wavelength characterizing the scattering wave function is small compared with the relevant dimensions of the scattering system.
The Wave-Optical Description Of Potential Scattering
Authors: Fröbrich P , Lipperheide R 
Published: November 2023

Abstract
In the preceding chapter we have discussed the semiclassical approximation to quantum scattering theory, where the dynamics is described in terms of classical trajectories. By analytic continuation of these trajectories into the complex plane the treatment has even been extended to the scattering by complex, i.e. absorptive, potentials.
Deep-inelastic collisions. Phenomena and theory
Authors: Fröbrich P , Lipperheide R 
Published: November 2023

Abstract
So far we have been concerned with direct and compound-nucleus reactions. We shall now consider another type of reaction, the dissipative or deep-inelastic collisions (DIC), which display features of both of the former. Deep-inelastic collisions are special to heavy ions, and are the more pronounced the heavier the collision partners. They can be described in classical terms, and generally take place at intermediate impact parameters. For large impact parameters they go over into the direct or quasi-elastic reactions, and for small impact parameters into fusion.
How to Give an LDS Formulation for a Logic
Authors: Gabbay Dov M 
Published: November 2023

Abstract
This chapter is concerned with the problem of how to give an LDS formulation into a traditionally presented logic. Given a logical system presented to us as a Hilbert system or as a Gentzen system or via tableaux or possibly via some algebraic semantics, we want to present it as an LDS system with a suitable algebra of labels. We realise there may be more than one way of doing it. We have to be more clear in what we mean by ‘presenting it’.
A General Theory of Structured Consequence Relations
Authors: Gabbay Dov M 
Published: November 2023

Abstract
In previous chapters we introduced a systematic framework where the logical units are labelled formulas (of the form t : A, where tis a term in some algebra) and where the logical rules are rules for manipulating both formulas and their labels. The intuition behind LDS is that in t : A, A carries declarative information and the label t, represents further information of a different nature which we do not want to ‘code’ into A. It turns out that many monotonic, non-monotonic and probabilistic systems can be presented as LDS systems. An LDS database Δ is a constellation of labelled formulas with additional structure on the labels.
Critical Points in an Algebra of Elementary Embeddings, II
Authors: Dougherty Randall 
Published: November 2023

Abstract
In this paper, we continue the study of a left-distributive algebra of elementary embeddings from the collection of sets of rank less than &gt;. to itself, as well as related finite left-distributive algebras (which can be defined without reference to large cardinals). In particular, we look at the critical points (least ordinals moved) of the elementary embeddings; simple statements about these ordinals can be reformulated as purely algebraic statements concerning the left distributive law. Previously, lower bounds on the number of critical points have been used to show that certain such algebraic statements, known to follow from large cardinals, require more than Primitive Recursive Arithmetic to prove. Here we present the first few steps of a program that, if it can be carried to completion, should give exact computations of the number of critical points, thereby showing that hypotheses only slightly beyond Primitive Recursive Arithmetic would suffice to prove the aforementioned algebraic statements.
Lukasiewicz Normal Forms and Torie Desingularizations
Authors: Mundici Daniele 
Published: November 2023

Abstract
Owing to its relations with AF C*-algebras, the infinite-valued calculus of Lukasiewicz is attracting renewed attention. A simple interpretation of non Boolean truth-values can be given in the context of Ulam’s game: this is the variant of the Twenty Questions game where the first player chooses a number x in a certain search space Sand the second player-the Investigator-must find x by asking a minimum number of Yes-No questions. In response the first player may lie up to ℓ times. Answers do not obey classical logic, because equal answers to the same repeated question are generally more informative than single answers. The current state of knowledge of the Investigator is a conjunction of answers.
Introduction
Authors: Molisch Andreas F , Oehry Bernhard P 
Published: November 2023

Abstract
Since the beginning of this century we have a rough understanding about the structure of atoms. They consist of a nucleus and electrons orbiting around this nucleus. We know that electrons occupy only certain discrete energy states. When they decay from a high-energy state to a low-energy state, they emit electromagnetic radiation. Usually, the low-energy state will be the ground state. In that case, the emitted radiation is called resonance radiation.1 Since the resonance photons have an energy that corresponds to the energy of the transition, they can easily be absorbed by other ground-state atoms. A ground-state atom that absorbs the resonance photon is excited to the upper state for some time and will eventually release the photon again. This absorption/reemission process can be repeated many times, until the photon finally escapes from the vapour. This process is called ‘radiation trapping’ (also known as ‘imprisonment of resonance radiation’, ‘radiative transfer of spectral lines’, ‘line transfer’, or ‘radiation diffusion’).
Mathematical Methods for the Holstein Equation
Authors: Molisch Andreas F , Oehry Bernhard P 
Published: November 2023

Abstract
In this chapter, we will discuss various methods for solving this equation. This chapter, as well as the following, is concentrated on the mathematical methods. The physical interpretation of the results will be given in Chapter 7. Holstein (1947) solved Eq. (5.1) by a variational technique. The symmetry of the Kernel function G(r, r’) = G(r’, r) permits this formulation of Eq. (5.1) as a variational problem.