id: 06471856
dt: b
an: 2015f.00094
au: Arnold, V. I.
ti: Experimental mathematics.
so: MSRI Mathematical Circles Library 16. Providence, RI: American Mathematical
Society (AMS); Berkeley, CA: Mathematical Sciences Research Institut
(MSRI) (ISBN 978-0-8218-9416-3/pbk). vii, 158~p. (2015).
py: 2015
pu: Providence, RI: American Mathematical Society (AMS); Berkeley, CA:
Mathematical Sciences Research Institut (MSRI)
la: EN
cc: A80 F60 H40 K20 I70
ut: combinatorial complexity; randomness; permutation; Young diagrams;
Frobenius numbers; additive semigroups; binary sequences; Galois fields
ci:
li:
ab: The work under review is indeed a delightful book full of ideas, deep
mathematical connections, unexpected solutions and open problems, all
consisting the portrait of a master of our time: Vladimir Arnold. These
collected lectures of Arnold at the Dubna summer camp of 2005 reveal a
very dynamic, effective, yet extremely demanding teaching. Well-stated
by Marc Saul at the preface : “The most exciting aspect of
mathematics for Arnold, seems to have been a dynamic search for pattern
through examination of many special cases". Thus, this book entitled
“Experimental Mathematics" achieves at each lecture an innovative
role of starting with “naive experimentation" and immediately after a
few pages, becoming extremely profound and condensed concerning
mathematical thinking. The book consists of four lectures and only
twenty references! Statistics of Topology and Algebra (Lecture 1),
Combinatorial Complexity and randomness (Lecture 2), Random
permutations and Young diagrams of their cycles (Lecture 3) and
Geometry of Frobenius numbers for additive semigroups (Lecture 4) form
a very competitive mathematical environment for each reader. Starting
from Hilbert’s sixteenth problem connected with smooth functions,
topology and algebraic geometry in Lecture 1, is passing to binary
sequences finally related with complexity and randomness in Galois
fields in Lecture 2. In Lecture 3 Young diagrams is the basis for
random permutations of larger numbers of elements, while in Lecture 4
Sylvester’s theorem leads to the geometry of continued fractions of
Frobenius numbers. It is not really four lectures, but a true mind
adventure in the world of “Experimental Mathematics" of Arnold.
rv: Panayiotis Vlamos (Athena)