bluuebunny/crossref_metadata_2025_split huggingface dataset:
The Fibonacci sequence, generalized Fibonacci sequences, and related topics
Authors: Cofman Judita 
Published: November 2023

Abstract
One of the most engaging puzzles in the history of mathematics was formulated by Leonardo of Pisa, nicknamed Fibonacci (1170-1250) in his celebrated book Liber abaci: How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?
The Chinese remainder theorem and invisible lattice points
Authors: Cofman Judita 
Published: November 2023

Abstract
The fourth-century Chinese mathematical text, titled Sun Tsu San Ching (Master Sun’s arithmetic manual), contains the following problem: There is an unknown number of objects. When counted in ‘threes’, the remainder is 2; when counted in ‘fives’, the remainder is 3; and when counted in ‘sevens’, the remainder is 2. How many objects are there? Sun Tsu describes a way for solving the problem and provides an answer to the question.
The Mfe Method Of Miller
Authors: Baines M J 
Published: November 2023

Abstract
The Moving Finite Element (MFE) method was introduced by Keith Miller (Miller and Miller 1981, Miller 1981) in order to obtain. numerical solutions of problems with steep moving fronts. The main ideas of the method are discussed in this chapter. The approach adopted is essentially that of Miller’s original papers since this appears to be the best way of introducing the subject. In later chapters however we shall describe other equivalent viewpoints which show the method in a different light.
Mfe In Higher Dimensions
Authors: Baines M J 
Published: November 2023

Abstract
One of the main attractions of the MFE method has been its ability to deal with problems in two and three dimensions on unstructured triangular and tetrahedral grids. In Section 1.4 of Chapter 1 we wrote down the weak forms (1.36) and (1.37) of the MFE residual minimization in any number of dimensions, leading to the matrix equation (1.40), namely,
Results From The Mfe Method
Authors: Baines M J 
Published: November 2023

Abstract
Although a full understanding of the workings of the MFE method is yet to come, this has not prevented the application of the method using penalty functions to a wide variety of problems by, among others, Gelinas et al. (1981), Mosher (1985), Hrymak et al. (1986), Dukowicz (1984), Juarez-Romero et al. (1988), Demkowicz (1984). The most prolific exponents of the method have been Miller and Carlson, using the global GWMFE formulation together with penalty functions and a stiff implicit solver for the time-stepping. Many other workers have followed a similar path, and between them these authors have covered a wide spectrum of practical problems, in some cases obtaining simulations not available by other numerical methods.
Best Fits With Adjustable Nodes
Authors: Baines M J 
Published: November 2023

Abstract
In this chapter we discuss the problem of finding optimal L fits to continuous functions with adjustable nodes via piecewise linear discontinuous functions. This gives the search an extra dimension, simplifying the conditions to be satisfied and linking strongly with the MFE procedure. Using a direct variational approach but with non-standard variations, interpreted numerically, new algorithms are devised, based on a two stage iteration process similar to that in Chapter 4, whose limit is the required best approximation. In this way we reduce the nonlinearity of the problem (virtually eliminating it for linear fits in one dimension) and obtain algorithms which are fast and robust. Using a similar approach, we also derive algorithms providing best piecewise constant L fits with adjustable nodes.
Genera And The Class Group
Authors: Rose H E 
Published: November 2023

Abstract
We begin by introducing the equivalence of forms modulo a prime (that is. studying forms locally; see Section 3 of Chapter 3), this gives a new class structure with classes called genera. Secondly, we will show that the set of equivalence classes (in the original sense) forms a group, called the class group, under the operation of composition of forms defined below. The genera are cosets of this group and this fact enables us to derive some arithmetical properties of the class group. Finally, we give a formula for h(d), the order of the class group for discriminant d.
The Prime Numbers
Authors: Rose H E 
Published: November 2023

Abstract
The study of the primes is one of the most important branches of number theory. The only other branch of similar importance is Diophantine equations and this will be considered in the final three chapters. The literature on the prime numbers is very extensive beginning with Pythagoras and Euclid; even so, many problems remain. There are two pre-eminent results: Dirichlet’s theorem and the prime number theorem (PNT).
Elliptic Curves: Further Results And Applications
Authors: Rose H E 
Published: November 2023

Abstract
In Chapter 3 we discussed the Hasse principle and noted that it does not apply to curves of genus L which includes the elliptic curves. Nevertheless, many properties of a curve C defined over a global field (Q for example) can be deduced from properties of C defined over a local field or a finite field. Most of this work is beyond the scope of this book, but we shall begin the study of curves defined over finite fields, define the corresponding group of ‘rational’ points, and give three applications.
Elasticity and anelasticity: continuous viewpoint*
Authors: Maugin Gerard A 
Published: November 2023

Abstract
The elastic behaviour of a material is characterized anthropomorphically by stating that it ‘remembers’ only one of its previous states, namely the natural, unstressed and unstrained state. Whether such a state had actually been occupied in the past or not, or whether it is a mere abstract notion which helps one to give an unambiguous mathematical definition, is another matter. In any case, whenever the ‘causes’ of motion and deformation are removed, the material must recover its original state and, therefore, such behaviour certainly does not exhibit thermodynamical irreversibility. Accordingly, all properties are derived from a potential, the elastic energy potential, usually taken as a function of strains.
Coupled fields in elasticity
Authors: Maugin Gerard A 
Published: November 2023

Abstract
In systems with electromagnetomechanical coupling, the electromagnetic component invariably gives rise to nonlinear effects or enhances them. The coupling with the electromagnetic component is also used to measure these effects. Some of the example of these systems are: (i) the elastic piezoelectric semiconductors, (ii) the already mentioned elasticferroelectrics of the molecular-group type and (iii) the elastic ferromagnets where nonlinearities could result from their semiconducting property, or large angular deviations, or even the up and down flipping of the molecular groups or the magnetic spins, respectively. Dispersion usually arises from the collective behaviour of sets of individual objects (such as magnetic spins) which are coupled through relatively long-range interactions. The ordering characteristics underlying several of these phases is essential in that respect.
Shock waves and phase-transition fronts in thermoelastic crystals
Authors: Maugin Gerard A 
Published: November 2023

Abstract
When structure is ignored, shock waves and phase-transition fronts both belong to the class of discontinuity fronts across which the first-order space-time derivatives of the basic fields undergo finite discontinuity jumps. This classification of discontinuity fronts goes back to celebrated studies by the mathematician Jacques Hadamard (1865-1963), who built on notions introduced by J. C. Maxwell (1831-1879), G. Riemann (1826--1866) and P. Duhem (1861-1916).
Reversible and non-dissipative processes
Authors: Ericksen J L 
Published: November 2023

Abstract
Of concern here are the various theories which are consistent with traditional and some newer views of thermodynamics, involving some specified set of admissible thermodynamic processes, with some prescription for calculating entropy changes and dissipation for these. Particularly, my aim is to encourage others to strive to clear up various questions related to the general theory of thermodynamic equilibrium, it being my experience that the literature on this is quite confused and confusing. There are various theories of equilibrium which involve considering some thermodynamic potential but with no explicit consideration of dissipation. For part of the discussion, we need to include these. Another kind of exception, closer to Albert Green‘s interests, will be mentioned in section 4. To focus our thoughts, it seems helpful to consider the following questions.
Convexity considerations for the biharmonic equation in plane polars with applications to elasticity
Authors: Flavin J N 
Published: November 2023

Abstract
Solutions of the biharmonic equation are considered in the arch-like region 0 &lt; θ &lt; α &lt; r &lt; b in the presence of boundary conditions ϕ = ϕ = 0 on the edges r = a, r = b ((r, θ) denoting plane polar coordinates). A cross-sectional measure F(θ) of the solution is considered and is proved to be convex in 0 for b/a ≤ exp π. If, additionally, the condition ϕ = ϕθ = 0 obtains on the edge 0 = α, F(θ) satisfies an enhanced inequality (generalized convexity); upper bounds for F(θ) in terms of suitable data follow.
Finite deformations of a fibre-reinforced cantilever: distributed-load solutions
Authors: England A H , Rogers T G , Bradford I D R 
Published: November 2023

Abstract
This paper considers the finite plane deformations of a cantilever beam composed of an elastic ideal fibre-reinforced material. The fibres are aligned along the beam. The cantilever is deformed by loads distributed over its lateral (upper and lower) surfaces. It is assumed that these forces either are ‘dead loads’ which do not change in magnitude or direction throughout the deformation, or that they act normally to the deformed lateral surfaces of the beam. In both cases solutions are derived using stationary-energy methods.
Explicit Q-functions
Authors: Hoffman P N , Humphreys J F 
Published: November 2023

Abstract
More details concerning the Q-functions are given in this chapter. Motivated by Theorem 7.21, we first define a function Q a for each integer sequence α; but then immediately show that if α: is not a strict partition, then Q a is either zero or is a multiple of Q A for a certain γ in 𝒟. The Q a are needed in Chapter 10, but they also play a role in a Laurent identity, (9.5) here. This is a generating “expression” for the Q a. It could be re-interpreted as a raising operator formula, such as occurs in the theory of Schur functions, but we prefer to use the Laurent formulation. It leads to four more formulae for Q A.
Construction of the Irreducible Negative Representations
Authors: Hoffman P N , Humphreys J F 
Published: November 2023

Abstract
In this chapter, we give a construction due to Nazarov (1989) of the irreducible negative representations of ˜. They are obtained from those of the group Ŝn (as defined in Remark (3) after Theorem 2.11) by multiplying the matrix for each generator sk by i to give the matrix for tk. We shall follow Nazarov closely, giving the matrices for Ŝn.
Combinatorial and Skew Q-functions
Authors: Hoffman P N , Humphreys J F 
Published: November 2023

Abstract
The first objective of this chapter is to find combinatorial descriptions of two sets of numbers: those which occur (i) as inner products &lt;q v, Q λ&gt;, and (ii) as coefficients in the expression of the product Q v Q μ in terms of the basis |Q λ|. The fom1er is essentially a special case of the latter, and leads to an enumerative description of the coefficients in the polynomial Qλ(X); in particular, these coefficients are non-negative integers. It also leads to a strengthening of the unitriangularity phenomena, in both the definition of Q λ, and in Theorem 8.6 relating the irreducibles &lt;λ&gt; to the induced representations ξλ. Schur’s fundamental orthogonality identity then yields an important enumerative result connecting the marked shifted tableau occurring in the above formulae with integer matrices which have “marked” entries.
The Shjffed Knuth Algorithm
Authors: Hoffman P N , Humphreys J F 
Published: November 2023

Abstract
This chapter will be devoted entirely to giving a second, purely combinatorial. proof of Theorem 12.11. We describe a number of algorithms and their inverses, and combine them to provide an explicit bijection. This material is the backbone of the combinatorial approach lo Q-functions. We shall make one further foray into this side of the subject, in the second half of the next chapter. The reader should consult the papers referred to in the notes to this and the next chapter for further details, as well as for alternative methods.
Introduction
Authors: Croarken Mary 
Published: November 2023

Abstract
In considering the centralization of scientific computation this book attempts to link together two areas of study: the prehistory of computers and the attitudes of scientists towards using machine methods of computation. More specifically, it examines how scientific computation was organized in Britain from the early 1900s to the mid-fifties. That is, from the time desk calculators began to become both readily available and reliable, to the invention and construction of the stored program computer. During this 50-year period the way in which computation was carried out changed significantly not only in terms of the technology used but also in the way in which computation was organized and financed. Advances in applied science meant that scientists and administrators had to find new ways of coping with an increasing demand for computation. The idea of centralizing computer power and providing user support developed slowly during the period. The concept of centralized compution culminated in 1945 with the creation of the National Physical Laboratory (NPL) Mathematics Division.
Background: aids to computation
Authors: Croarken Mary 
Published: November 2023

Abstract
The most commonly used computational aids available in the 1900s were mathematical tables, slide rules, nomograms, mathematical instruments, and desk calculating machines. Of these only the mathematical tables and the desk calculators lent themselves to the type of large-scale repetitive calculations which could be centralized with advantage. The increasing use of desk calculating machines during the early twentieth century laid the foundation for computing centres to become a possibility but nevertheless mathematical tables, slide rules, and mathematical instruments had an important role to play.
The National Physical Laboratory Mathematics Division: a national computing centre
Authors: Croarken Mary 
Published: November 2023

Abstract
During 1946 the NPL Mathematics Division became fully operational. By the end of that year the Division had a staff of 50 divided into five sections: the General Computing Section, the Punched Card Section, the Differential Analyser Section, the Statistics Section, and the Automatic Computing Engine (ACE) Section. The Mathematics Division was operating more or less within the guidelines set out in the 1944 DSIR Interdepartmental Committee Report. From the description of a Central Mathematics Station as outlined in that report it might be assumed that the NPL Mathematics Division also embodied the concept of a computing centre defined by the characteristics given in Chapter l. However, before such a conclusion can be reached it is important to examine the NPL Mathematics Division more closely.
Theory and practice of tern poral logic programming
Authors: Orgun Mehmet A , Wadge* William W 
Published: November 2023

Abstract
Chronolog is a logic programming language based on temporal logic and possible worlds semantics. Temporal logic allows us to describe dynamic and time-dependent properties of certain problems in a direct and problem-oriented way. In Chronolog programs, users specify time-varying predicates; thus the notion of dynamic change and dataflow-style of non-terminating computations can be naturally specified in Chronolog. The declarative semantics of Chronolog programs can be characterized in terms of the least model semantics; hence Chronolog does not have some of the problems associated with non-logical extensions of logic programming. Chronolog extended with choice predicates (arbitrary single-valued relations) is suitable for modelling mutual exclusion problems such as that of the dining philosophers. A natural extension of Chronolog with modules allows us to specify objects with internal memory.
A promenade from provability to consistency
Authors: * Philippe Balbiani 
Published: November 2023

Abstract
The semantics of logic programming with negation is addressed by means of modal logic. Inspired by the well-known three-valued semantics, a modal completion formula of logic programs is defined with the help of modal operators. Depending on the logical be haviour of these operators, the modal formula completion will alternatively possess the soundness and completeness properties with respect to ‘SLDNF-refutability’ or the soundness and completeness properties with respect to ‘non-SLDNF-refutability’. It will allow a logical characterization of some properties possessed by SLDNF proof trees as well. Lastly, since one of these properties is closely related to the closed-world assumption rule, the completion formula will give an almost complete proof procedure for this non-monotonic rule of inference.
Selected papers
Authors: Hogger Christopher John 
Published: November 2023

Abstract
The following papers are worth reading for their lucid exposition, foundational importance a,,d historical significance.
"A machine-oriented logic based on the resolution principle" by J. A. Robinson. Journal of the ACM 12, 1965, pp.23--41.
"Computation and deduction" by P. J. Hayes.
Proc. of the 2nd Symposium on the Mathematical Foundations of Computer Science, Czechoslovak Academy of Sciences, 1973, pp. l 05-118.
"Predicate logic as a programming language" by R. A. Kowalski.
Proc. of IFIP-74, North Holland Publishing Company, Amsterdam, 1974, pp. 569-574.
The alleged deficiencies
Authors: Hogger Christopher John 
Published: November 2023

Abstract
Fifteen years ago logic programming was barely visible in the computing landscape. But during the ten years that followed its acclaim spread far and wide, and many of those working in the field felt there was quite a good chance that it would substantially displace older and supposedly less satisfactory formalisms. The Japanese had invested heavily in it for the purposes of their Fifth Generation Computer Systems Project, and the number of world-wide sites in which Prolog was used numbered tens-perhaps even hundreds--of thousands. There was a lot of interest in expert systems, many of which were written in Prolog. Industries in the UK and elsewhere in Europe readily joined with academia in various programmes of funded research and development, and numerous companies came into being as suppliers of logic programming technology.
More model theory definitions
Authors: Hogger Christopher John 
Published: November 2023

Abstract
An interpretation which gives the value true to a sentence is said to satisfy
the sentence; such an interpretation is called a model for the sentence.
Example-the interpretation first chosen in Theme 11 is a model for the sentence likes(chris, mummy).
An interpretation.which does not satisfy a sentence is called a counter¬ model for that sentence.
By extension, we also have the notion of a model (or counter-model) for a set of sentences: an interpretation is a model for the set if and only if it is a model for each of the set’s members.
A sentence which has at least one model is said to be satisfiable.
A sentence which has no models is said to be unsatisfiable.
Example- likes(chris, mummy) is satisfiable.
Moving literals within clauses
Authors: Hogger Christopher John 
Published: November 2023

Abstract
The classical representation of a clause is a disjunction of literals. However, any such clause can be rewritten, with no loss of equivalence, as a sentence in conditional format having the form disjunction of literals if conjunction of literals Thus a clause such as A or B or ,C or ,D or ,E can be rewritten, for example, in this conditional format or B if C&D&E
Starting with the classical format, we obtained this new format by disjoining, on the left of an ‘if’, all the positive literals and by conjoining, on the right of the ‘if’, all the atomic parts of the negative literals. That the two versions are equivalent is owed solely to the logical interpretations of the connectives:
Computations and answers
Authors: Hogger Christopher John 
Published: November 2023

Abstract
Various species of derivation may evolve from a query at the root of a search tree. Let us take another look at the tree obtained using the ‘append’ program for the query
In any such tree, a derivation beginning with the root query (Ql) is called a computation from that query. In general, a tree may contain both finite computations and infinite computations (‘loops’).
The unification algorithm
Authors: Hogger Christopher John 
Published: November 2023

Abstract
There are many different species of algorithms available for computing most¬ general unifiers. The one most commonly employed in logic program interpreters is known as Robinson’s Algorithm (named after the discoverer of resolution).Its task is as follows. Given as input two atoms of the form
in which the ri and ti are any terms, its aim is to decide whether the atoms are unifiable and-if they are-to deliver as output their most-general unifier; if they are not unifiable then the output is the message “failure”.
In the initial set-up, all the term-pairs &lt; ri, ti &gt; are stored on a stack S. Another stack 0, initially empty, is made available for accommodating those bindings (called ‘replacements’ in earlier Themes) which the algorithm will construct in the course of comparing each pair of terms on S. These two stacks are the only data structures required.
Varying the selection rules
Authors: Hogger Christopher John 
Published: November 2023

Abstract
Given a query and a program, the choice of computation rule determines a corresponding SLD-tree. Assuming that the search procedure is top-down and depth-first, the choice of search rule determines only the order in which the tree’s computations are generated. Below we sketch some SLD-trees generated from a simple path-finding problem (seeking paths commencing at node ‘a’).
Incremental construction of the minimal model
Authors: Hogger Christopher John 
Published: November 2023

Abstract
There is a simple incremental method for constructing the minimal model of any definite program P. This method is not strictly an algorithm since, in the case where MM(P) is infinite, it may not complete the construction in a finite number of steps; however, the more steps we perform, the better the approximation.
The method starts by choosing some interpretation I 1&lt;;;;B(P). Later on we shall see that there is a preferred choice of I1 and that not all choices are adequate. Choosing I1 can be viewed as guessing which atoms have to be true in any model of P. Now consider any clause in G(P)