Spending symmetry.
Terence TaoIn preparation
Last updated: Nov 18, 2012
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.
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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).) [...]