monograph
A monograph is a specialist work of writing (in contrast to reference works) or exhibition on a single subject or an aspect of a subject, often by a single author or artist, and usually on a scholarly subject.
In library cataloging, ''monograph ...
Marian Pour-El
Marian Boykan Pour-El (April 29, 1928 – June 10, 2009) was an American mathematical logician who did pioneering work in computable analysis.
Early life and education
Marian Boykan was born in 1928 in New York City; her parents were dentist Josep ...
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. 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.
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, the remaining two parts of the book concerns the authors' results. 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. The authors' main tools include the notions of a ''computability structure'', a pair of a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
and an axiomatically-characterized set of its sequences, and of an ''effective generating set'', a member of the set of sequences whose
linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
is dense in the space.
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. However, they show that this phenomenon cannot occur for the
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
or for
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
.
The book also includes a collection of open problems, likely to inspire its readers to more research in this area.
Audience and reception
The book is self-contained, and targeted at researchers in mathematical analysis and computability; reviewers Douglas Bridges and Robin Gandy disagree over which of these two groups it is better aimed at. Although co-author
Marian Pour-El
Marian Boykan Pour-El (April 29, 1928 – June 10, 2009) was an American mathematical logician who did pioneering work in computable analysis.
Early life and education
Marian Boykan was born in 1928 in New York City; her parents were dentist Josep ...
came from a background in mathematical logic, and the two series in which the book was published both have logic in their title, readers are not expected to be familiar with logic.
Despite complaining about the formality of the presentation and that the authors did not aim to include all recent developments in computable analysis, reviewer
Rod Downey
Rodney Graham Downey (born 20 September 1957) is a New Zealand and Australian mathematician and computer scientist,. a professor in the School of Mathematics and Statistics at Victoria University of Wellington in New Zealand.
References
{{reflist, refs=
{{citation
, last = Downey , first = Rodney G. , authorlink = Rod Downey
, journal =
Mathematical Reviews
''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science.
The AMS also pu ...