Computability in Analysis and Physics

Computability in Analysis and Physics is a monograph on computable analysis by Marian Pour-El and J. Ian Richards. It was published by Springer-Verlag in their Perspectives in Mathematical Logic series in 1989, and reprinted by the Association for Symbolic Logic and Cambridge University Press in their Perspectives in Logic series in 2016.

The book concerns computable analysis, a branch of mathematical analysis founded by Alan Turing and concerned with the computability of constructions in analysis. This area is connected to, but distinct from, constructive analysis, reverse mathematics, and numerical analysis. The early development of the field was summarized in a book by Oliver Aberth, Computable Analysis (1980), and Computability in Analysis and Physics provides an update, incorporating substantial developments in this area by its authors.^[1] In contrast to the Russian school of computable analysis led by Andrey Markov Jr., it views computability as a distinguishing property of mathematical objects among others, rather than developing a theory that concerns only computable objects.^[2]

After an initial section of the book, introducing computable analysis and leading up to an example of John Myhill of a computable continuously differentiable function whose derivative is not computable,^[1] the remaining two parts of the book concerns the authors' results.^[3] These include the results that, for a computable self-adjoint operator, the eigenvalues are individually computable, but their sequence is (in general) not; the existence of a computable self-adjoint operator for which 0 is an eigenvalue of multiplicity one with no computable eigenvectors; and the equivalence of computability and boundedness for operators.^[1] The authors' main tools include the notions of a computability structure, a pair of a Banach space and an axiomatically characterized set of its sequences, and of an effective generating set, a member of the set of sequences whose linear span is dense in the space.^[3][4]

The authors are motivated in part by the computability of solutions to differential equations. They provide an example of computable and continuous initial conditions for the wave equation (with however a non-computable gradient) that lead to a continuous but not computable solution at a later time.^[3][4] However, they show that this phenomenon cannot occur for the heat equation or for Laplace's equation.^[2]

The book also includes a collection of open problems,^[2][4] likely to inspire its readers to more research in this area.^[3]

El libro es autosuficiente y está dirigido a investigadores en análisis matemático y computabilidad; ^{[ 1 ]} los revisores Douglas Bridges y Robin Gandy discrepan sobre a cuál de estos dos grupos está mejor dirigido. ^{[ 3 ] [ 4 ]} Aunque la coautora Marian Pour-El tenía formación en lógica matemática , y las dos series en las que se publicó el libro incluyen la palabra «lógica» en su título, no se espera que los lectores estén familiarizados con la lógica. ^{[ 2 ]}

A pesar de quejarse de la formalidad de la presentación y de que los autores no pretendían incluir todos los avances recientes en análisis computable, el crítico Rod Downey escribe que este libro "es claramente imprescindible para cualquiera cuya investigación se centre en esta área" ^{[ 1 ]} y Gandy lo califica como "un libro interesante, legible y muy bien escrito" ^{[ 4 ] .}