Spending symmetry.
Terence TaoIn preparation
Last updated: Nov 19, 2021
This continues my series of books derived from my blog. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, “An epsilon of room“, “An introduction to measure theory“, “Topics in random matrix theory“, “Higher order Fourier analysis“, “Compactness and contradiction“, and “Hilbert’s fifth problem and related topics“.
A draft version of the MS can be found here.
Pre-errata (to be corrected in the published version):
- Page ix: the parenthetical reference to “compactness and contradiction” should be deleted.
- Page 34: On line 6, “Let now analyse” should be “Let us now analyse”.
- Page 45: In Section 2.2.5, one should use the tensor product instead of the direct sum , making the necessary changes to the formulae (e.g. replacing by ).
- Page 55: In line 4 of the statement of Theorem 4.1.1: “exactly one of the following statements hold” should be “exactly one of the following statements holds”.
- Page 89: “Converge almost everywhere” should be “Converges almost everywhere”.
- Page 144: On the third line before Conjecture 7.2.4, a space is missing after [BoSo1978].
- Page 199: should be .
- Page 208: In Lemma 8.5.1 and its proof, one should replace with a measurable superset , since might not itself be measurable.
- Page 233: The references [Ba1967,Ba1967b] were puvblished in 1967 rather than 1966; also [BoSaZi2011] has been publishhed.
- Page 235: The reference [Er1979] should be Erdős, Paul Some unconventional problems in number theory. Math. Mag. 52 (1979), no. 2, 67–70. The reference [Ka1940] should refer to Erdős instead of Erdos.
Thanks to Alan Chang, Nick Gill, Gerry Myerson, Aditya Guha Roy, Ho Boon Suan, and Po Lam Yung for corrections.
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18 November, 2012 at 10:16 am
Spending symmetry « What’s new
[…] my 2011 blog posts (and also my Google buzzes and Google+ posts from that year), entitled “Spending symmetry“. The PDF of this draft is available here. This is again a rather assorted (and […]
7 December, 2012 at 12:43 am
arminniakan
Wish even more success for you, Professor.
I cannot wait to come to the deparment to study and hopefully become your student Someday.
29 December, 2012 at 1:04 pm
A mathematical formalisation of dimensional analysis « What’s new
[…] We have already observed that to verify a dimensionally consistent statement between dimensionful quantities, it suffices to do so for a single choice of the dimension parameters ; one can view this as being analogous to the transfer principle in nonstandard analysis, relating dimensionful mathematics with dimensionless mathematics. Thus, for instance, if have the units of , , and respectively, then to verify the dimensionally consistent identity , it suffices to do so for a single choice of units . For instance, one can choose a set of units (such as Planck units) for which the speed of light becomes , in which case the dimensionally consistent identity simplifies to the dimensionally inconsistent identity . Note that once we sacrifice dimensional consistency, though, we cannot then transfer back to the dimensionful setting; the identity does not hold for all choices of units, only the special choice of units for which . So we see a tradeoff between the freedom to vary units, and the freedom to work with dimensionally inconsistent equations; one can spend one freedom for another, but one cannot have both at the same time. (This is closely related to the concept of spending symmetry, which I discuss for instance in this post (or in Section 2.1 of this book).) […]
15 May, 2020 at 9:06 am
Nick Gill
Just in case you’re looking for errors: The preface refers to “the titular COMPACTNESS AND CONTRADICTION” which I think is an erroneous reference to a previous book in the series. (Or maybe I’ve misunderstood something.)
[Thanks, this is now added to the pre-errata. -T]
14 July, 2020 at 10:54 pm
autoditactics
Google+ is also now defunct, and the link in the preface is a 404.
12 December, 2021 at 8:32 pm
Aditya Guha Roy
On page 235: in reference [Ka1940] Paul Erdos should be corrected to Paul Erdős.
[Correction added, thanks – T.]
12 December, 2021 at 8:42 pm
Aditya Guha Roy
I really like your text a lot.
There are several beautiful applications of symmetry in analyzing random walks.
I mention two of them below, I hope you would like them:
1. Probabilities of chess pieces visiting certain squares before others.
(For chess pieces the movements are very symmetric, for instance for the rook the probability of visiting each of the corners from some cell in the inner 7 by 7 part of the board are the same.)
2. (This one is by prof. Laszlo Lovasz and prof. Peter Winkler)
Suppose you do a random walk on a n cycle, travelling at each step 1 unit either in the clockwise or the anticlockwise direction each with probability 1/2, and stopping as soon as you have visited all the vertices. Then the vertex W where you stop is uniformly distributed on the set of n-1 vertices (excluding the one where you started, since that is considered visited right from the beginning).