1,649,430 research outputs found
Descending Dungeons and Iterated Base-Changing
For real numbers a, b> 1, let as a_b denote the result of interpreting a in
base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to
be numbers of the form a_b_c_d_..._e, parenthesized either from the bottom
upwards (preferred) or from the top downwards. Among other things, we show that
the sequences of dungeons with n-th terms 10_11_12_..._(n-1)_n or
n_(n-1)_..._12_11_10 grow roughly like 10^{10^{n log log n}}, where the
logarithms are to the base 10. We also investigate the behavior as n increases
of the sequence a_a_a_..._a, with n a's, parenthesized from the bottom upwards.
This converges either to a single number (e.g. to the golden ratio if a = 1.1),
to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a =
frac{100{99).Comment: 11 pages; new version takes into account comments from referees;
version of Sep 25 2007 inculdes a new theorem and several small improvement
Seven Staggering Sequences
When my "Handbook of Integer Sequences" came out in 1973, Philip Morrison
gave it an enthusiastic review in the Scientific American and Martin Gardner
was kind enough to say in his Mathematical Games column that "every
recreational mathematician should buy a copy forthwith." That book contained
2372 sequences. Today the "On-Line Encyclopedia of Integer Sequences" contains
117000 sequences. This paper will describe seven that I find especially
interesting. These are the EKG sequence, Gijswijt's sequence, a numerical
analog of Aronson's sequence, approximate squaring, the integrality of n-th
roots of generating functions, dissections, and the kissing number problem.
(Paper for conference in honor of Martin Gardner's 91st birthday.)Comment: 12 pages. A somewhat different version appeared in "Homage to a Pied
Puzzler", E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters,
Wellesley, MA, 2009, pp. 93-11
Decorous lower bounds for minimum linear arrangement
Minimum Linear Arrangement is a classical basic combinatorial optimization problem from the 1960s, which turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which the best known solution value often has one more digit than the best known lower bound value. In this paper, we propose a linear-programming based approach to compute lower bounds on the optimum. This allows us, for the first time, to show that the best known solutions are indeed not far from optimal for most of the benchmark instances
- …