I suggest removing this paragraph (unfortunately this will affect page breaks). I don’t think this paragraph is correct, and the rest of the section doesn’t seem to reference the information contained in this paragraph.

Computer processors implement many functions that are not (implemented in terms of) polynomials over any field, such as saturation arithmetic, operations on n-bit words (eg AND, OR, XOR, number of bits set to 1, bit shifting, etc), various comparisons, bit packing/unpacking, integer/floating point conversions, and many others. See for example here:

https://en.wikipedia.org/wiki/X86_instruction_listings

The information that functions like logarithms are computed in terms of polynomials is also not entirely correct. Taylor series expansion is one of the techniques but even then it is only part of the algorithm. For more details, see:

http://math.stackexchange.com/questions/61209/what-algorithm-is-used-by-computers-to-calculate-logarithms

totaly agree, but it may seems that the next step update the old one(i, ii, iii), it it definitely = 2(A(1,1)=2 as state in the example he gave ), it is not an error if in sequential programming language(like an update of the previous statement), he should explaint it anyway.

]]>112.1 lines -7 and -8

(i) and (ii) clash in defining A(1,1) as first 3 then 2.

]]>But I think that I am going to love the book anyway. ]]>

I’m glad to say that that one was picked up (see post).

]]>Greatly enjoying dipping into this volume. I’ve noticed one or two errors in the Lie Theory article (pp.229-2234) as follows:

231.1, line16: The Lie algebras u(n) and su(n) are not the same: su(n) has the extra trace 0 condition.

231.2, para 3: The proposed motivational calculation for the Lie bracket doesn’t work. In fact you have to start with (at least) the degree two Taylor approximations to A and B, ie. let A=I+eX+e^2X^2/2 etc in order for the commutator calculation to work out as claimed. This needs changing in a future edition, as it would confuse a non-expert who tries to check it. In fact, for this reason, it might be better to change the order of exposition in this section a bit, so as to introduce the idea of the exponential first, and then come back to the Lie bracket.

232.1, line 11: It’s incorrect that the exponential map is surjective for a general connected group. It’s true if the group is also compact (standard theorem), but false eg. for the connected group SL(2,R) (see Sepanski: Compact Lie Groups, p.87). What is true, of course, is that the image of the exponential generates the group.

233.1, line -3: whose n+1 coordinates …

Finally, in several places (230.2, 234.1) the notation Sp(2n) is used for the compact symplectic group. Of course, this is just a matter of notation, but most people would denote this as Sp(n) in analogy with O(n) and U(n) (Sepanski, Knapp, Brocker-tom Dieck, Chevalley, Fulton-Harris…, but admittedly not Bump).

These are just minor quibbles. This is a great book!

Graham Williams

]]>Actually the sentence is correct as stated. “Special” here doesn’t mean “remarkable” so much as “particular”. The idea is that these equations look too specific to be of general significance, but in fact that isn’t the case at all.

]]>For example, equations of the form y^2 = f(x), where f(x) is a cubic polynomial in x, may NOT look rather special, but in fact the ELLIPTIC CURVES [III.21] that they define are central to modern number theory, including the proof of Fermat’s last theorem.

]]>Yes — it has been for quite some time.

]]>1) p156 An explicite exemple off “a^b= irra. is a = squareroot(2) , b = log base 2 of 9; where a^b=3” (notice that page 223 tell us that a^b = irra. when a = b =squareroot(2).

2) p 169.1 both denominators are (1+x^2)

3) p 171 not 1/4,-1/4,3/4,-3/4, 4/3,-4/3 but 1/4,-1/4, 2/3,-2/3,3/2,-3/2

4) p309.2 last line Z^2 should be R^2

5) p558 line 4 Québecoise should be Québécoise (I am one of the Labelle “frères”)

6)p771 2xT(2) should be 2xT(x)

7) 1902 Longeur should be Longueur; 1901 Absolut should be Absolu

Thx again.

]]>(Article IV.7 contains a correct statement for n=4, and a nice description of why the proof in high dimensions turned out to be easier than in dimensions 3 and 4, but doesn’t contain the statement for general n.)

One way to correct V.25 would be to write `a compact simply-connected smooth n-manifold whose homology groups are isomorphic to those of an n-sphere must be homeomorphic to an n-sphere’. In fact it suffices to know that the mth homology group is the same as that of an n-sphere for each m <= n/2, but this would complicate the statement even more than introducing homology groups.

Another way to correct it would be to write `a compact smooth n-manifold M with the property that for each m<n, any map from the m-sphere S^m to M extends to a map from the (m+1)-ball to M

must be homeomorphic to an n-sphere'. Again, it suffices to know this for each m with m <= n/2. (This is the statement given in IV.7 in the case n=4.) This statement could be shortened by referring to `homotopy groups' as defined in IV.6.2, but the longer and more elementary statement is probably better.

A third way would be to write `a compact smooth n-manifold which is homotopy equivalent to the n-sphere must be homeomorphic to the n-sphere', referring to IV.6.2 for the definition of a homotopy equivalence. This one is my preference, for what it's worth.

]]>(this is about 645.1 -1, not 646.1 -1)

]]>And thank you for this terrific book!

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