Structure and Randomness: pages from year one of a mathematical blog
American Mathematical Society, 2008
298 pages
ISBN-10: 0-8218-4695-7
ISBN-13: 978-0-8218-4695-7
Last updated Sep 26, 2020
This is a book version of my blog, covering many (though not all) of my articles in 2007, reworked into a publishable format (and in particular, with formal and updated references).
A draft copy of the book is available here. Note that the formatting for this internet version is significantly different from that in the final print version, in particular the page numbering does not correspond at all.
Here is a (2MB) PDF file containing the front cover of the book.
The official AMS web page for this book is here.
A review of the book for the American Scientist is here.
Here is a review for the MAA.
— Errata —
- Page 21: In Proposition 1.8,
should be
.
- Page 22: In Proposition 1.10,
should be
.Page 51: In Lemma 1.34(2),
should be
.
- Page 54: The phrase “nonstandard ultrafilter on
” requires some clarification. It should be “ultrafilter on
that extends an ultraproduct of nonprincipal ultrafilters on
“.
- Page 86: In the final display, the exponent of
on the LHS should be deleted. Similarly on the first display of page 87.
- Page 95, Figure 1: The graph of
is missing two edges (namely, the long diagonal edges between opposite corners of the graph).
- Page 96: The inequality
is only valid for connected graphs with at least one cycle. In general, one only has
. One must then delete all terms involving the number 6 in the rest of the section. One can then replace “To solve the optimisation problem exactly, one needs to solve a cubic; but we can perform a much cheaper computation” by “One can solve the optimisation problem exactly, but we can perform a cheaper computation”.
- Page 101: Footnote 67 is incorrect and should be deleted.
- Page 102:
should be
.
- Page 103: The sentence “More generally, any Riemannian manifold…” is incorrect and should be deleted, replaced instead with the sentence “Similarly, the hy0erbolic plane
is isomorphic to
“.
- Page 104: In first paragraph: the modular curve should be
. In the footnote, add “and an obvious left action of
,” after the final comma.
- Page 128: In the second and third displays,
should be
.
- Page 129: “decaying faster” should be “decaying faster or having smaller amplitude”.
- Page 143: The proof that
is not correct as stated (the perturbations indicated do not preserve the property of being an
-flow). A correct argument is as follows. Call an edge of an
-flow unsaturated if it has weight strictly between 0 and
, and similarly call a vertex unsaturated if its net inflow or outflow is strictly less than
. Observe that if e is an unsaturated edge, then the final vertex u of e will either have an unsaturated edge leading out of it (if u is unsaturated) or another unsaturated edge leading into it (if u is saturated). Similarly, the initial vertex u’ of e will either have an unsaturated edge leading into it (if u is unsaturated) or another unsaturated edge leading out of it (if u is saturated). Thus, if there is at least one unsaturated edge, then by iterating the above observations, one can find an oriented cycle along unsaturated edges with the property that at any saturated vertex u, the number of edges flowing along the cycle into u equals the number of edges flowing against the cycle into u, and the number of edges flowing along the cycle out of u equals the number of edges flowing against the cycle out of u. For this cycle, one can modify the flow as indicated in the text to reduce the number of edges in the
-flow.
- Page 137: In the last line of Case 1, “multiply (1.47) by…” should be “raise (1.47) to the power
and then multiply by…”, and
should be
.
- Page 148: In the first line,
should read
.
- Page 180: In the second display,
should be
.
- Page 257: When selecting the large prime
, it is necessary that
is larger than 2. This is needed in order for the binomial expansion of
to be expressible in the desired form
where the polynomials
have coefficients in the p-adic integers, because the number of times
divides
can be bounded below by
, which goes to infinity as
provided that
is larger than 2.
- Page 267: In the third display,
should be
.
- Section 3.10: In (2.8),
should be
.
Thanks to Ravindra Bapat, Alan Chang, Dion, Norman Hardy, Matthew Kahle, Chris Kimmel, JamesL, Avi Levy, Russ Lyons, Arturo Magadin, Kirane Mokhtar, David P. Moulton, Kamil Rychlewicz , David Speyer, Nikodem Szpak, Tobias and Tony, Chao Weng, Sheng-Peng Wu, and Yuncheng for corrections.
11 comments
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4 February, 2008 at 6:00 am
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2 May, 2012 at 7:32 am
Arturo Magidin
I believe there is an error on page 95, Figure 1. The graph labeled as $K_{3,3}$ is not $K_{3,3}$, since the corner vertices are not connected to the opposite corners.
[Correction added, thanks – T.]
2 August, 2013 at 6:18 pm
Matt
I recently purchased this book in e-form through Google books – I was surprised that unlike the beautifully published print version, which I also own, Google appears to be selling a rather low-quality scanned version of your book, wrinkles visible, which is not searchable, text-reflow-able, nor available to read when not plugged into the internet. If I know mathematicians, I know that most would be appalled that their work was being presented, even sold, in such degraded form. Would you be kind enough to contact your publisher, AMS, and urge them to supply EPUB versions of your books to Google? This would be greatly appreciated, and I very much look forward to reading your articles on the subway without needing a wheelbarrow to carry the all the dead trees :) Thanks!
26 May, 2016 at 8:55 am
Kerri
Structure and randomness : where a simple law explodes into quantum elements of order that describe the bounds of physics. Where space is as important in understanding the order of structures as energy and matter is. These concepts are Real within Prime structure and pi, some of our favorite explorations of infinite possibility.
13 November, 2017 at 12:48 pm
Anonymous
There are a couple errors on pages 21-22. Propositions 1.8 and 1.10 are not correctly stated; a counterexample to both is
\epsilon = 4/5
k = 2
M = 4
x_1 = 0, x_2 = 0.1, x_3 = 0.9, x_4 = 1
The bound needed on M in Proposition 1.8 is stronger: $\lfloor\frac{M-1}{k}\rfloor \geq \epsilon$. Along the same lines, the bound needed on M in Proposition 1.10 is $\lfloor\log_2 (M) \rfloor$.
With these modifications, the proofs alluded to after each proposition do work. I suggest also changing $x_k, x_2k, x_3k, …$ to $x_1, x_{1+k}, x_{1+2k}, …$ on page 21 and removing the variable k from Proposition 1.10.
Thank you for the excellent article; I’m still working through it.
[Errata added, thanks – T.]
2 October, 2019 at 5:16 am
rongjiang pan
A Proof of Collatz Conjecture in Binary
https://github.com/ronaldpan/Collatz