bluuebunny/crossref_metadata_2025_split huggingface dataset:
A universal system and its applications
Authors: Smullyan Raymond M 
Published: November 2023

Abstract
We now wish to construct a “universal” system (U) in which we can, so to speak, express all propositions of the form that such and such a number is in such and such a recursively enumerable set, or more generally that such and such an n-tuple of numbers is in such and such a recursively enumerable relation. A complete knowledge of which sentences in this system are true, and which are not, would imply a complete knowledge of all formalized mathematics. But we will see that the problem is undecidable — a result that Post[13] calls “Gödel’s Theorem in Miniature.” Also, our universal system will be the basis for two fundamental results in recursion theory — the Enumeration Theorem and the Iteration Theorem, which are indispensable for many of the chapters that follow.
Actions count
Authors: Neumann Peter M , Stoy Gabrielle A , THOMPSON EDWARD C 
Published: November 2023

Abstract
In this chapter we discuss a famous counting theorem that is the foundation for a large area of combinatorics known as enumeration theory. Throughout the chapter n will be a finite set and G a finite group acting on 0. For g e G let
Thus fixΩ(g) consists of those elements of Ω that are in cycles of length 1 in the permutation induced by g; with the conventional rule that cycles of length 1 are not recorded, fixΩ(g) consists of those points of O that do not appear in the cycle notation for g. We write OrbΩ(G) for the set of G-orbits inn. With this notation the theorem is:
Euclidean geometry
Authors: Neumann Peter M , Stoy Gabrielle A , Thompson Edward C 
Published: November 2023

Abstract
It will be noted that the ratio of the (essentially positive) Euclidean distances only gives the absolute value of the invariant (δac/δab). It is this fact which gives rise to some tiresome technical difficulties in classical Euclidean geometry, which relies principally on the notion of distance. In a non- algebraic development the notion of distance has to be supported by a notion of ‘order’ or ‘between-ness’ or ‘separation’ of points on a line.
Belief Revision
Authors: Gardenfors Peter , Rott Hans 
Published: November 2023

Abstract
Now suppose that, as a matter of fact, the bird caught in the trap turns out to be black. This means that you want to add the fact -&gt;C, i.e., the negation of C, to the database. But then the database becomes inconsistent. If you want to keep the database consistent—which is normally a sound methodology—you need to revise it. This means that some of the beliefs in the original database must be retracted. You don’t want to give up all of the beliefs since this would be an unnecessary loss of valuable information. So you have to choose between retracting A, B1, B1 -&gt; B, or A A B C. When giving up a belief you have to decide which of the reasons for this belief to retain and which to retract (this is the backward direction).
Modal Deduction with Applications in Epistemic and Temporal Logics
Authors: Cerro Luis Farinas del , Herzig Andreas 
Published: November 2023

Abstract
Abstract
The widespread use of non-classical logics in Artificial Intelligence makes it desirable to have automated deduction methods (and in particular resolution methods) for these logics at our disposal. In this chapter we present several automated deduction methods for modal logics. We focus on the problems of normal forming and variable skolemization, which prove to be harder than in classical logic because of the enriched language.
Euclidean geometry
Authors: Roe John 
Published: November 2023

Abstract
Let S be an affine space. We saw in the last chapter that if we also assume that S has the extra structure of an absolute distance |AB| between two points A and B, then it is possible to define a dot product u • v between any two vectors u and V in S. This dot product has the four properties of linearity, symmetry, positivity, and definiteness, which are summarized in Figure 4.1.
Solid geometry
Authors: Roe John 
Published: November 2023

Abstract
In this chapter we will use vector methods to study the geometry of three dimensions, traditionally known as solid geometry. As we did for plane geometry, we will begin by discussing the concept of orientation. In two dimensions, an orientation specifies a ‘positive direction of rotation’ (6.1.2), and thereby divides the possible coordinate systems into right-handed and left handed. In three dimensions, we will find that an orientation makes possible a similar distinction between right-handed and left-handed coordinate systems.
Differential geometry of curves
Authors: Roe John 
Published: November 2023

Abstract
Most of the geometry that we have been studying so far has been what is called algebraic geometry. Algebraic geometry studies curves and surfaces that are represented by polynomial equations — such as the equation x2 + y1 = 1 of the circle — using only algebraic facts about polynomials and their roots. For instance, we discussed the tangent lines to a conic in terms of double roots of the equation for the intersection of a line and a conic. This is a typical kind of algebraic-geometry argument.
Iterative Methods for Linear Systems
Authors: Elman Howard C 
Published: November 2023

Abstract
In Section 2, we introduce the basic ideas of stationary iterative methods and consider several particular examples of these, namely the Jacobi, Gauss- Seidel, SOR, and SSOR methods. We outline some results on the convergence of them, for both general matrices and those with special structure. In Section 3, we give an overview of Krylov subspace methods for systems where the coefficient matrix is symmetric.
Hierarchical Preconditioners for Elliptic Partial Differential Equations
Authors: Bank Randolph E 
Published: November 2023

Abstract
In this chapter we shall present an introductory survey of the hierarchical basis preconditioners for elliptic partial differential equations. In the past decade this class of methods has drawn much interest. Hierarchical basis methods, [4], [5], [19], [20], and [25], are closely related to multigrid methods [16], [17], and [22], and to the BPX class of methods, [8], [9], [23], [27], and [28). They share with them a rapid rate of convergence, although hierarchical basis methods have a logarithmic-like dependence on the size of the problem, whereas multigrid and BPX have no h-dependence and are essentially optimal in their rates of convergence, [21] and [7].
Hyperbolic Systems
Authors: Pandolfi Maurizio 
Published: November 2023

Abstract
Extended summary of the contribution given at the SERC Numerical Analysis Summer School, Lancaster University, July 1992.) A large variety of physical phenomena is described by hyperbolic systems. Fluid dynamics is probably the field of major relevance: compressible flows (unsteady and steady supersonic), shallow waters, hydraulic transients in ducts, etc. Also, the propagation of stresses in particular structures in space applications or through underground layers is governed by hyperbolic systems. Phenomena in astrophysics such as explosions and implosions or solar winds are described by these systems, as well as many further examples in physics and technology.
Introduction
Authors: Huxley M N 
Published: November 2023

Abstract
This book is written from the standpoint of analytic number theory, a subject whose name may seem contradictory. Number theory deals with the behaviour of the integers under addition and multiplication, and analysis with continuity and limit operations. The logicians have told us that any problem in mathematics can be disguised as finding integer solutions of a set of simultaneous quadratic equations. This means that we must be prepared to use any method that comes to hand. We may classify problems in number theory only by the methods used, not by the form in which they are stated. An argument is algebraic if it uses only addition and multiplication, and analytic if it uses the ordering of the integers, or the ordering of the rational numbers within the real line, which is a consequence of the ordering of the integers. From this point of view ‘elementary number theory’ is not a third category, but a description of the early levels. It is in this sense that the first part of this book is ‘elementary’.
The rational points close to a curve
Authors: Huxley M N 
Published: November 2023

Abstract
with denominators in a range Q ≤q &lt;2Q. We do not require that the highest common factor of a, b and q be one, but that each rational point that can be written as (a/q,b/q) with Q ≤ q &lt; 2Q is counted once only, however many triples of integers a, b, q give the required ratios. We can consider the triple a, b, q as a point of the projective plane, although the curve y =f(x) need not be algebraic. This is the same as counting triples (a,b,q) of integers satisfying (4.1.1) with 1 ≤ q &lt; 2Q and highest common factor (a, b, q) = 1.
Lattice points and area
Authors: Huxley M N 
Published: November 2023

Abstract
The analytic method for estimating the discrepancy between the number of lattice points inside a closed curve involves dividing the curve into five or more pieces, taking local coordinates y =f(x) on each piece, and estimating the contribution ∑ p(f(m)) that each piece of the curve makes to the discrepancy. The classical estimate is Corollary 2 to Lemma 5.4.3. We use the Fourier expansion of Lemma 5.3.1, which leads to the sums considered in Chapter 8. Our first task is to use the estimates for the First Spacing Problem in Chapter 13 and the Second Spacing Problem in Chapter 16 to complete the estimation of the exponential sum, and deduce the bounds for the rounding error sum. As in the last section, we require that certain derivatives and determinants of derivatives do not vanish. We shall show that the local coordinate systems can be chosen to satisfy these conditions.
Sums with modular form coefficients
Authors: Huxley M N 
Published: November 2023

Abstract
Remarks The second term in (20.1.7) appears when we pass from a smoothed sum in Lemma 10.3.1 (with N≍MT -1/3) to the sum (20.1.1). This term may be reduced by appealing to a stronger bound than Lemma 10.2.1, such as the full result of Rankin (1939) or the results of Moreno and Shahidi (1983) (Lemma 10.3.3) or Deligne (1974) (Lemma 10.3.2).
Polar Decomposition And Gauss Decomposition
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
Every real positive definite symmetric matrix x can be diagonalized, that is, there exist an orthogonal matrix u and a positive diagonal matrix d such that x = udu′. This gives a decomposition of x which we call the polar decomposition because of its similarity to polar coordinates in the plane. Another decomposition of x is the Gauss decomposition x = tt′, where t is a lower triangular matrix with positive diagonal entries. In this chapter we describe the generalizations of these decompositions to arbitrary Euclidean Jordan algebras. The main results are the formulas of Theorem VI.2.3 and Theorem Vl.3.9, which express the integral of a function on Ω in terms of the polar decomposition and the Gauss decomposition.
Tube Domains Over Convex Cones
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
In this chapter we study complex tube domains over proper convex cones. Our results are ultimately based on the simple fact that the Laplace transform of an integrable function supported in a proper cone defines a holomorphic function in the tube domain over the dual cone. After recalling some basic notions we give a complete exposition of the theory of the Bergman kernel and Bergman metric on general complex domains. Then we proceed to the explicit discussion of the Bergman and Hardy spaces of holomorphic functions on a tube domain over a proper convex cone. We also discuss the Bergman-Shilov boundary and the associated Poisson kernel for such domains.
Function Spaces On Symmetric Domains Of Tube Type
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
In the first three sections of this chapter we briefly consider certain Hilbert spaces of holomorphic functions which form a natural generalization of the Bergman space. They are of interest from the viewpoint of representation theory, since the group G(D) has natural unitary representations acting on them. We do not give a very exhaustive discussion of these results since they are special cases of results for arbitrary bounded symmetric domains. In the final section we discuss Poisson transforms of functions defined on the Shilov boundary and we prove that they satisfy the so called Hua differential equations. The results of this section are special to the tube-type case; they do not extend in a simple way to the general case.
Representations Of Jordan Algebras And Euclidean Fourier Analysis
Authors: Faraut Jacques , Korányi Adam 
Published: November 2023

Abstract
Some questions in classical analysis can be considered as related to representations of Jordan algebras. We will study some of them in this chapter: Wishart distributions, Hankel transform, generalized K-Bessel function, and some zeta integrals. Let V be a simple Euclidean Jordan algebra of rank r, degree d and dimension n. Let ϕ be a self-adjoint representation of V on a Euclidean vector space E of dimension N, and Q be the associated V-valued quadratic form on E (see Section 4 of Chapter IV).
Solution techniques of partial differential equations
Authors: Rahman Matiur 
Published: November 2023

Abstract
We saw in the previous chapter that the motion of a fluid particle can be described by partial differential equations. In the study of potential wave theory, partial differential equations play a central role. We have seen, for example, how the velocity potential and the stream function, in fluid flow, are described by Laplace’s equation. In the following chapters the main topics will be gravity waves and tidal waves. Here again, the physics of the problems will be described by a set of partial differential equations. Thus, for the proper study of the above phenomena, it is imperative that we master the solution techniques of partial differential equations.
Fundamental results on critical points
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
Every multiple zero of a polynomial is a critical point. How are the other critical points located relative to the set of zeros? An answer is given by the following theorem, which generalizes the above-mentioned fact about the critical points of a polynomial whose zeros are collinear (see Fig. 2.1). In spite of its simple proof, this theorem will turn out to be of central importance in the analytic theory of polynomials. It is not only a key to various results on critical points, but also has numerous applications of other kinds.
Coefficient estimates
Authors: Rahman Q I , Schmeisser G 
Published: November 2023

Abstract
These inequalities have obvious interpretations for the coefficients of trigonometric polynomials. Real trigonometric polynomials and those which are non-negative deserve to be treated separately. However, in view of Proposition 12.4.5, any estimate for the coefficients of real trigonometric polynomials has a counterpart for the coefficients of self-inversive polynomials. In addition, certain estimates for the coefficients of non-negative trigonometric polynomials, when combined with Lemma 12.2.4, lead us to interesting inequalities for the coefficients of polynomials. Due to its special significance, we shall also determine the sharp upper bound for the modulus of the sum of the coefficients of a sine polynomial.
Sampling by Generalized Kernels
Authors: Stens R L 
Published: November 2023

Abstract
There are several possible extensions of this theorem. For example, one may use other than rectangular lattices, which in general are more efficient (see e.g. Peterson and Middleton 1962; Mersereau 1979; Mersereau and Speake 1983; Stark 1979; Higgins 1985; Butzer and Hinsen 1989; Higgins 1996, Chapter 14). However, in any case, bandlimitation is a rather restrictive condition, since a band-limited function is in particular an entire function which cannot be simultaneously duration-limited, in view of the uniqueness theorem for analytic functions.
The ASD Yang-Mills equation
Authors: Mason L J , Woodhouse N M J 
Published: November 2023

Abstract
In this chapter, we shall look at various ways of expressing the ASD condition, first as a system of first-order equations on the components of the gauge potential, and then as the commutativity condition on a Lax pair of operators. We shall then look at two second-order forms: the first involves a matrix potential that has become known as Yang’s J-matrix, although in the Euclidean and ultrahyperbolic cases, it has an older interpretation as a Hermitian metric on a holomorphic vector bundle; the second involves another second-order potential, which we call the K-matrix. The resulting partial differential equations will be the central objects of study in the rest of the book. Both Yang’s equation and the K-matrix equation are derived from Lagrangians, and therefore their solution spaces have symplectic structures. In the first section, we set the scene by looking at the corresponding equations in Maxwell’s theory.
Reduction to two dimensions
Authors: Mason L J , Woodhouse N M J 
Published: November 2023

Abstract
We now come to the central examples of reduction, in which the symmetry group has two-dimensional orbits in space-time and the reduced system has two independent variables. First we shall look at the two-dimensional translation groups, from which we obtain the ASD equations on a Riemann surface, the chiral equations, harmonic map equations, the KdV and NLS equations, and various parts of the nKdV hierarchies. When we add additional discrete symmetries, we get the extended Toda field equation, other cases of the harmonic condition on maps to homogeneous spaces, and the n-wave equation. Second, we shall consider other two-dimensional groups that act freely on space-time. A notable example is the reduction to the Ernst equation in general relativity, where the group is generated by a translation and a rotation. We also consider the possibility that the symmetry group has higher dimension, but does not act freely. An obvious example is the rotation group SO(3), acting on hyperplanes in lE. Here the nonsingular orbits are 2-spheres, but the group is three dimensional. One reduction by this symmetry is Liouville’s equation. Another example is the complex Euclidean group, which has two-dimensional orbits with nontrivial isotropy. Here the ASDYM equation reduces to the Toda field equation. 1
The Fourier Transform And Adhm Construction
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
We have not yet met any interesting examples of solutions to the anti-self-dual (ASD) equations. In the present chapter we will remedy this and describe geometric constructions for ASD connections over two special four-manifolds: the flat four-torus and round (conformally flat) four-sphere. For connections over the four-torus we define a ‘Fourier transform’ interchanging ASD solutions on different bundles. These results are of a somewhat theoretical interest since they do not immediately yield any explicit solutions; but we will then go on to apply the same ideas to derive the ‘ADHM construction’ for connections over S4(which we regard as the conformal compactification of R’). This gives a complete and concrete description of all solutions in terms of certain matrix data, and the results will provide a prototype for the general theory of Chapter 4.
Excision And Gluing
Authors: Donaldson S k. , Kronheimer P B 
Published: November 2023

Abstract
This chapter brings together a number of loosely related topics from analysis, the general context being the description of solutions to differential equations, depending on a parameter, for limiting values of the parameter. We begin by considering the excision principle for the index of linear elliptic operators. As we shall explain below, this principle leads rapidly to the proof of the vital index formula given in Chapter 4 for the virtual dimension of Yang-Mills moduli spaces. We shall show that, as an alternative to the standard proof using pseudo-differential operators, one can prove this principle by introducing a suitable deformation of the differential operator. This discussion sets the scene for the more specialized geometrical topics considered in the rest of the chapter.
Degenerative Parameters
Authors: Dwork Bernard 
Published: November 2023

Abstract
The kernel and images of π|u|χu:Wa+u→W^a are stable under the connection and so Theorem 6.9.1 provides a source of values of parameters a for which the differential eqn (5.7) becomes reducible over the field of rational functions. Such values of parameters provide opportunities for comparison with the classical literature. We shall list some examples based on the calculations of Chapter 12 and shall check Conjecture 6.3.1 in a few cases. The factor π will sometimes be disregarded in this chapter.
Existence: Recursive Methods
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
Often, when referring to the direct product, we mean the generalized direct product. An important special case occurs when |W| = 3, and μ= 1. This is a tripling constructionthat forms a TS(3ν. λ)from a TS(ν λ) and an arbitrary quasigroup of order ν(or a latin square of order ν,or a TD(3, ν)).
Independent Sets
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
A subset S ⊆V in a TS(v, λ) (V,B)is independentif B ⊈Sfor all BE B.An independent set is maximalif, for all x∈ V\ S, S⋃ ( x) is not independent (that is, it is maximal with respect to set inclusion). It is maximumif it has the largest cardinality of any independent set in the design. Independent sets arc often called arcs,and maximal independent sets are called complete arcs.
Nested and Derived Triple Systems
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
In this chapter, we examine extensions of triple systems to designs with larger block size. We have already seen a number of problems involving the placement of the blocks of a triple system (or partial triple system) into a ‘larger design’. For example, embedding permits us to increase the number of elements, immersing increases the index, and enclosing increases both. Here we consider operations that increase the block size.
Directed Triple Systems
Authors: Colbourn Charles J , Rosa Alexander 
Published: November 2023

Abstract
In this and the next chapter, we examine extemions of designs (particularly triple systems), in which blocks are taken to contain orderedrather than unordered pairs. Vieprimarily consider two different types of ordered blocks. The transitive triple(a,b,c) is said to contain the ordered pairs (a,b),(a,c) and (b,c).The cyclic triple(a, b,c)contains the ordered pairs (a,b), (b,c)and (c, a). We examine directed triple systems in which the blocks are transitive triples, Mendelsohn triple systems in which the blocks are cyclic triples, and hybrid triple systems in which the blocks are permitted to be either.