Event trees, and more generally, event spaces, can be used to provide a foundation for mathematical probability that includes a systematic understanding of causality. This foundation justifies the use of statistics in...
ISBN:
(纸本)0821806114
Event trees, and more generally, event spaces, can be used to provide a foundation for mathematical probability that includes a systematic understanding of causality. This foundation justifies the use of statistics in causal investigation and provides a rigorous semantics for causal reasoning.
This paper is intended to give an introduction to several techniques used in automated theorem proving, and based on the concept of ordering on terms and more generally on formulas. Different orderings and their autom...
ISBN:
(纸本)0821806114
This paper is intended to give an introduction to several techniques used in automated theorem proving, and based on the concept of ordering on terms and more generally on formulas. Different orderings and their automation are presented and several applications are considered. We show how orderings provide a way to incorporate equality decision procedures in theorem provers. We also illustrate their fundamental role to automatise proofs by induction and proofs by refutation. Finally, we explain how they allow pruning the search space of a theorem prover by defining strategies to restrict application of inference rules.
In this paper, we present the mathematical foundations of constraint programming. A program is a set of constraints, i.e. a mathematical formula. A set of constraints is viewed as an implicit representation of the set...
ISBN:
(纸本)0821806114
In this paper, we present the mathematical foundations of constraint programming. A program is a set of constraints, i.e. a mathematical formula. A set of constraints is viewed as an implicit representation of the set of all constraints that it entails. There is a query system such that an answer to a query is a relationship that is satisfied if and only if the query is entailed by the system. And most importantly there exists a SINGLE algorithm to answer all queries (an oracle). In the case of logic programming, the constraint domain is Horn clauses and the algorithm is resolution. The soundness and completeness of resolution guarantees a well-defined programming system. In the domain of arithmetic constraints, the algorithm is quantifier elimination, and Tarski's theorem provides the counterpart to the soundness and completeness of resolution. As the cost of implementation in the general case is prohibitive, we restrict ourselves to the linear case. A program is a conjunction of linear constraints which represents a polyhedral set. Fourier's algorithm, the counterpart of Tarski's for the linear case, provides the parametric query answering mechanism. Interestingly, in this framework, classic linear programming results are interpreted in a new light. In particular, duality is given a quantifier elimination interpretation rather than the usual economic one, and a link between the simplex algorithm and the ubiquitous Helly's theorem is shown. Note that proofs are not included in the following presentation. The interested reader is referred to the literature.
This book introduces readers to key ideas and applications of computational algebraic geometry. Beginning with the discovery of Gröbner bases and fueled by the advent of modern computers and the rediscovery of re...
ISBN:
(纸本)9780821807507
This book introduces readers to key ideas and applications of computational algebraic geometry. Beginning with the discovery of Gröbner bases and fueled by the advent of modern computers and the rediscovery of resultants, computational algebraic geometry has grown rapidly in importance. The fact that “crunching equations” is now as easy as “crunching numbers” has had a profound impact in recent years. At the same time, the mathematics used in computational algebraic geometry is unusually elegant and accessible, which makes the subject easy to learn and easy to apply. This book begins with an introduction to Gröbner bases and resultants, then discusses some of the more recent methods for solving systems of polynomial equations. A sampler of possible applications follows, including computer-aided geometric design, complex information systems, integer programming, and algebraic coding theory. The lectures in the book assume no previous acquaintance with the material.
There exists a history of great expectations and large investments involving Artificial Intelligence (AI). There are also notable shortfalls and memorable disappointments. One major controversy regarding AI is just h...
There exists a history of great expectations and large investments involving Artificial Intelligence (AI). There are also notable shortfalls and memorable disappointments. One major controversy regarding AI is just how mathematical a field it is or should be. This text includes contributions that examine the connections between AI and mathematics, demonstrating the potential for mathematical applications and exposing some of the more mathematical areas within AI. The goal is to stimulate interest in people who can contribute to the field or use its results. Included is work by M. Newborn on the famous Deep Blue chess match. He discusses highly mathematical techniques involving graph theory, combinatorics and probability and statistics. G. Shafer offers his development of probability through probability trees with some of the results appearing here for the first time. M. Golumbic treats temporal reasoning with ties to the famous Frame Problem. His contribution involves logic, combinatorics and graph theory and leads to two chapters with logical themes. H. Kirchner explains how ordering techniques in automated reasoning systems make deduction more efficient. Constraint logic programming is discussed by C. Lassez, who shows its intimate ties to linear programming with crucial theorems going back to Fourier. V. Nalwa's work provides a brief tour of computer vision, tying it to mathematics—from combinatorics, probability and geometry to partial differential equations. All authors are gifted expositors and are current contributors to the field. The wide scope of the volume includes research problems, research tools and good motivational material for teaching.
This book is the result of an AMS Short Course on Knots and Physics that was held in San Francisco (January 1994). The range of the course went beyond knots to the study of invariants of low dimensional manifolds and ...
ISBN:
(纸本)0821803808
This book is the result of an AMS Short Course on Knots and Physics that was held in San Francisco (January 1994). The range of the course went beyond knots to the study of invariants of low dimensional manifolds and extensions of this work to four manifolds and to higher dimensions. The authors use ideas and methods of mathematical physics to extract topological information about knots and manifolds. Features: A basic introduction to knot polynomials in relation to statistical link invariants. Concise introductions to topological quantum field theories and to the role of knot theory in quantum gravity. Knots and Physics would be an excellent supplement to a course on algebraic topology or a physics course on field theory.
This book connects coding theory with actual applications in consumer electronics and with other areas of mathematics. Different Aspects of Coding Theory covers in detail the mathematical foundations of digital data s...
ISBN:
(纸本)9780821803790
This book connects coding theory with actual applications in consumer electronics and with other areas of mathematics. Different Aspects of Coding Theory covers in detail the mathematical foundations of digital data storage and makes connections to symbolic dynamics, linear systems, and finite automata. It also explores the use of algebraic geometry within coding theory and examines links with finite geometry, statistics, and theoretical computer science. Features: A unique combination of mathematical theory and engineering practice. Much diversity and variety among chapters, thus offering broad appeal. Topics relevant to mathematicians, statisticians, engineers, and computer scientists. Contributions by recognized scholars.
Geometry and topology are subjects generally considered to be “pure” mathematics. Recently, however, some of the methods and results in these two areas have found new utility in both wet-lab science (biology and che...
ISBN:
(纸本)9780821855027
Geometry and topology are subjects generally considered to be “pure” mathematics. Recently, however, some of the methods and results in these two areas have found new utility in both wet-lab science (biology and chemistry) and theoretical physics. Conversely, science is influencing mathematics, from posing questions that call for the construction of mathematical models to exporting theoretical methods of attack on long-standing problems of mathematical interest. Based on an AMS Short Course held in January 1992, this book contains six introductory articles on these intriguing new connections. There are articles by a chemist and a biologist about mathematics, and four articles by mathematicians writing about science. All are expository and require no specific knowledge of the science and mathematics involved. Because this book communicates the excitement and utility of mathematics research at an elementary level, it is an excellent textbook in an advanced undergraduate mathematics course.
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