twjudson / aata Goto Github PK

This is pretty minor, but worth mentioning if it hadn't been before. At https://github.com/twjudson/aata/blob/3f8983e52b996228a67dc3f6d624d3baaff61cdc/src/permute.xml#L1586 it looks like in context (such as the hint) like $D_{2n}$ is defined as the dihedral group of the $n$-gon, but earlier it looks like the notation is clearly $D_n$. Anyway, it's confusing to ask for the center of $D_n$ when all the numbers given are evens.

Likely throughout the book, but specifically in the proof of 1.3.1 and 2 you've invoked the idempotence of or without explicitly stating it. It might not be a bad idea to include a quick boolean algebra recap so the logic used for the proofs in the book is entirely self-contained.

As a student, one of the things that helps me make sure I have actually proved something is to label every single step I take in a proof with the name of the theorem or definition that lets me take that step.

the statement x or x = x may very well seem self-evident, but when you're first learning it's really good to learn not to trust yourself to make tat judgement and rely instead on a previously named definition or proven theorem. Just giving me a name for the thing that lets me go from {x: x e A or x e A} to {x: x e A} lets me be sure I can actually do it instead of being just pretty sure it sounds right.

Wikipedia's got a pretty good set of basic logic properties, and it wouldn't take too much space to just introduce them without proof as a reference:
Associativity of or: x or (y or z) = (x or y) or z
Associativity of and: x and (y and z) = (x and y) and z
Commutativity of or: x or y = y or x
Commutativity of and: x and y = y and x
Distributivity of and over or: x and (y or z) = (x and y) or (x and z)
Distributivity of or over and: x or (y and z) = (x or y) and (x or z)
Identity of or: x or false = x
Identity of and: x and true = x
Annihilator for or: x or true = true
Annihilator for and: x and false = false
Idempotence of or: x or x = x
Idempotence of and: x and x = x
Absorption: x or (x and y) = x = x and (x or y)

Also it'd be some fun foreshadowing for some of the concepts and terms that are to come. Wait, logic itself is a ring with which iv'e been working with all along!? No waaaaay!

The definition of function in Chapter 1 reads

We will define a mapping or function <m>f \subset A \times B</m> from a set <m>A</m> to a set <m>B</m> to be the special type of relation where <m>(a, b) \in f</m> if for every element <m>a \in A</m> there exists a unique element <m>b \in B</m>.

This seems a little scrambled. Should this be something like

the special type of relation in which for each element <m>a \in A</m> there is a unique element <m>b \in B</m> such that <m>(a, b) \in f</m>.

as in the 2014 edition?

Messing up a braille testing app for @zorkow:

http://abstract.ups.edu/aata/ssets-ection-sets-and-equivalence-relations.html

I suspect an xml:id with typos. Could be worth cruising HTML output for any more?

In the index, there is one instance of Sylow p-subgroup that needs a space between "Sylow" and "p".

Potential typo in Section 5.2 of the book. In the proof of Theorem 5.21, you mention that if an element of D_n sends 1 to k and then 2 to k-1, then the element is sr^k. But isn’t it actually r^ks? Notice that unless n is even and k=n/2, then sr^k does not send 1 to k.

Hi,

Thanks a lot for this open source textbook.

Algebraic coding theory appears to be a dependency for chapters 9 and beyond in the
dependency chart in the preface. I would think that it should be an individual branch, such
as chapters 6 and 7.

Is that indeed an omission or is there a good reason for that?

Thanks a lot

In line 67 of struct.xml, the index tag seems to be incorrect. In the html, it appears as

Groups!p-groups

As question 1 is currently worded, my students don't quite know what to do. May I suggest something like:

1. Complete-the-square to solve the general quadratic equation

ax^2+bx+c=0,

and in this process, derive the "quadratic formula" ....

Apologies if this is a misunderstanding by me, but exercise 21.5.1(d) seems unreasonably difficult (or perhaps surprisingly so for a first exercise). If I understand correctly (from other books), the minimal polynomial of primitive roots of unity are the cyclotomic polynomials. Figuring that out and proving those are irreducible (when n is not necessarily prime) is hard.

Also in Chapter 7, but in other places that deal with code words, can you add a command for all the binary code words, such as 111, 000, etc.? Especially in aligned displays and in tables, I switch to a tabular numeral so they all have the same width and line up. Rob provides a font hook for tables, but unfortunately we don't want all tables to do this, and there are many places outside of tables where they occur.

Dear authors / editors,

2016 version of AATA including Sage Exercises.
Regarding section 3.7, subsection Groups of symmetries:

In the final Cayley table for the section, rows and columns of the table are named in the following order id u1 u3 r1 r2 u2. I believe there is a typo here. Look for instance at an example: applying the rotation r1 twice, per the table yields the reflection u3. This cannot be true, it should yield r2.

A fix should be fast, I believe the correct order of permutation names to correspond to the table output by Sage is as follows:

#Build the Cayley table
aata_names = ['id', 'u3', 'r1', 'r2', 'u1', 'u2']
triangle.cayley_table(names=aata_names)

Thank you for a great book!

Best regards,
Janus

Numbers relating to math that aren't in math mode end up in the oldstyle numerals. I'm sure I don't have a complete list (perhaps there would be some easy way to search the XML?). For example, in figures 1.7, 1.9, 5.22, 5.23, 19.7, examples 4.5, 4.6, exercise 7.2.c.

A good test of alternate/additional MBX indices (not ready yet) would be an index of Sage commands, distinct from other index entries.

Exercise 11.5.1 (in Additional Exercises) id homomorph-exercise-aut-G

Do you want to say the group operation is composition? Just to be clear what the automorphism group is exactly.

Problem has \leq where maybe you want \subseteq or similar?

Should $G$ be finite if you want an automorphism to be construed as a permutation?

Theorem 23.7: The proof assumes that the degree of the splitting field E of f(x) is the same the degree of the polynomial f(x). The main idea is correct, but to count G(E/F) I also found it necessary (but maybe here I'm being dense) to consider cosets of G(E/F(\alpha)), viewed as a subgroup of G(E/F). Comment from N. Touikan.

id in math doesn't use \identity in a few places (though most do):
in text before example 1.16
in the proof of theorem 1.20
in the notation list
in figure 3.6

Solutions of Section 1.3, Question 25a. The solution reads "NThe" instead of "The".

In the “Historical Note” in chapter 3 (Groups), there is a missing space between “polynomials’” and “coefficients”.

In the file finite.xml, shouldn't line 76 read

The polynomial x^2 - 2 is separable over {\mathbb Q(\sqrt{2})} since it factors...

\mathbb contains the polynomial coefficients, but the field extension \mathbb Q(\sqrt{2}) contains the roots \pm \sqrt{2}...

Several questionable spots in the text. The page numbers come from the 2019 edition.

1.   Page 214, exercise 32.  This is trivially true, as R is a subring of itself.  If we restrict R’ to be a proper subring, then the claim is no longer true, so I’m not sure what this question is getting at.
   

2.   Page 253.  The term nonzero is not defined when discussing the definition of an atom.  Presumably it means non-O.  The first definition of an atom does not make sense: An element a \in B is an atom of B if a \neq O and a \wedge b = a for all nonzero b \in B.  In the Boolean algebra that is the divisors of 26 together with “divides” as the ordering, 2 and 13 should both be atoms, but 2 \wedge 13 = 1 = O, implying neither are atoms. 
   

3.   Page 258, exercise 4.  This question is faulty.  The set of divisors of n together with “divides” as the ordering is a Boolean algebra if and only if n is square-free.  Thus, n = 36 does not form a Boolean algebra.
   

4.   Page 258, exercise 9.  The problem reads \mathcal{P}(X)=2^n, when it should read |\mathcal{P}(X)|=2^n.
   

Who knew?

But in Lars-Daniel Öhman's article in the Intelligencer from last year, he points out that while many (most?) texts (including mine) state this, it is false. Relative to Peano's first four axioms, the induction axiom and the well-ordering principle are not equivalent axioms. In particular, there is an easy to describe model for well-ordering where (finite) induction does not hold - the set of ordinals up to $\omega+\omega$.

Since AATA is explicitly mentioned as one of the texts the author makes clear he "would recommend to ... anyone", I figured I should open an issue once I had time 😄 here is the line:

I would suggest a commit, but find this is the sort of thing that an author would want to keep in his or her individual style, as opposed to fixing a typo.

In the PDF of the 2020 edition, the reference for footnote 2 is on page 67, but its content appears on the next page. This is somewhat confusing because it makes it look like the "2" reference on page 67 is a typo.

Refs commit 2088bb9, issue #35

This link returns a 404: http://abstract.pugetsound.edu/aata/images/cover_aata_2019.png.

The second sentence of this proof doesn't read correctly. I think sentence fragments were deleted inadvertently. I suggest it should be split into two sentences and read something like:

"If |G|=p then clearly G itself is the required subgroup. We now assume that every group of order k, where p <= k < n and p divides k, has an element of order p."

This symbol is first used in chapter 6: Fermat's little theorem.
IMHO this barred symbol could deserve an explanatory note or an additional sentence, as it might be a bit confusing to the reader.
(While I knew about its opposite division symbol '|', as a student in France I've never seen this \notdivide symbol in any maths course from secondary school to CS engineering).

I can create a PR if necessary.

I found a number of figures and tables within other environments, such as examples. Because the examples use a square at the end, this created some odd pagebreaks and other issues. Since the figures are referred to by number and page anyway, how would you feel about moving them just after the other environments?

Lemma 15.6 says:

The proof is given like this

:
It would be cleaner, I think, to point out that the stabilizer of K is N(K) n H and then invoke Theorem 14.11 which says
that the size of an orbit is the index of the stabilizer.

There are instances of GAP, CPU, RSA, AES, DES that don't use \acronym{} (though many do).

Section 14.2 states that computing the number of partitions of n into positive integers is NP-complete, but I think there may be some confusion with the partition problem (which asks whether positive integers a_1,...,a_n can be partitioned into two sets with equal sums). Computing p(n) is not NP-complete; Johansson (https://arxiv.org/abs/1205.5991) implemented an algorithm that uses the Hardy-Ramanujan-Rademacher formula to calculate the partition function p(n) in time roughly sqrt(n). Since the number of digits in p(n) is on the order of sqrt(n), this is softly optimal.

Very early, we have a proof of the Division Algorithm, where it looks like two cases in the proof are being structured as p with em being used to make titles.

<em>Existence of <m>q</m> and <m>r</m>.</em>

It is doing poorly in a conversion to braille. We now have a cases element for this situation.

I didn't look to see if there are more. It'd be good to fix as many as possible.

Exercise 29 in Chapter 5 asks about D_8, D_{10}, and D_n. https://github.com/twjudson/aata/blob/master/src/permute.xml#L1559

But is it asking for the center for any n (so, even/odd cases, but strange that the examples only ask for even) or for only even n (so, notational ambiguity)? In either case, it should be clarified somewhat.

Chapter 7, many code words are given a \mono{} font command, but not earlier in the chapter (for example, DOJHEUD and ALGEBRA).

Is "cycle structure" used in exercise 5.11 before it is defined in exercise 6.15?

Braille conversion is exposing some figures without captions. Numbers are from August 2022 PDF posted online. At least three are commutative diagrams. Braille version (all versions?) would be improved with an informative caption for each.

Figure 12.25
Figure 21.33
Figure 21.35
Figure 23.8

Consider:

If G acts on X on the left, and you want a left action of G on Hom(X,Y), then you need an inverse here:
\sigma(f) = f\circ\sigma^{-1}. Otherwise it's a right action, in which case sigma(f) is confusing. I think there's no way to avoid
being more explicit here, one way or another.

On the web version, in the second paragraph of section 13.1, there is an extra "\index{Generators for a group}" presumably left over from the LaTeX version.

Also, in the sentence before Lemma 13.9, "If" should be "It"

When images were split into two (to live in a sidebyside) the @permid tended to be duplicated (on image and at least once on figure), when it is meant to be unique. I suspect you are suppose to leave them off entirely for a change like this, and then new ones will be generated as part of declaring a new edition. @davidfarmer can advise.

The first run of 21 EPUB validation errors should indicate all the duplicates (Duplicate). I suspect the later ones are consequernces and will go away.

I find some triplicates when I search because the old source was "commented out". This makes diagnosis harder, and we can always recover old source by rolling back with git, which is very easy and not dangerous.

I can try building EPUB again, if you put a notice here that the duplicates are gone and changes pushed.

ERROR(RSC-005): aata.epub/EPUB/package.opf(1,9042): Error while parsing file: Duplicate "leO"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,9158): Error while parsing file: Duplicate "leO"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,17744): Error while parsing file: Duplicate "dFT"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,17840): Error while parsing file: Duplicate "dFT"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,17934): Error while parsing file: Duplicate "JNc"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,18028): Error while parsing file: Duplicate "JNc"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,18135): Error while parsing file: Duplicate "pUl"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,18242): Error while parsing file: Duplicate "pUl"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,18490): Error while parsing file: Duplicate "Wbu"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,18599): Error while parsing file: Duplicate "Wbu"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,19101): Error while parsing file: Duplicate "aLn"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,19200): Error while parsing file: Duplicate "aLn"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,22331): Error while parsing file: Duplicate "sZv"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,22447): Error while parsing file: Duplicate "sZv"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,30209): Error while parsing file: Duplicate "eUq"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,30310): Error while parsing file: Duplicate "eUq"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,36517): Error while parsing file: Duplicate "OAV"
ERROR(RSC-005): aata.epub/EPUB/package.opf(1,36620): Error while parsing file: Duplicate "OAV"
ERROR(RSC-005): aata.epub/EPUB/xhtml/boolean-section-algebra-of-circuits.xhtml(84,57781): Error while parsing file: Duplicate ID "YgP"
ERROR(RSC-005): aata.epub/EPUB/xhtml/boolean-section-algebra-of-circuits.xhtml(84,60917): Error while parsing file: Duplicate ID "YgP"
ERROR(RSC-008): aata.epub/EPUB/xhtml/permute-section-dihedral-groups.xhtml(84,207105): Referenced resource "EPUB/xhtml/generated/latex-image/permute-transpositions-cube-left.svg" is not declared in the OPF manifest.
ERROR(RSC-008): aata.epub/EPUB/xhtml/matrix-section-groups.xhtml(84,394624): Referenced resource "EPUB/xhtml/generated/latex-image/matrix-sl2r-a.svg" is not declared in the OPF manifest.
ERROR(RSC-008): aata.epub/EPUB/xhtml/matrix-section-groups.xhtml(84,739144): Referenced resource "EPUB/xhtml/generated/latex-image/matrix-o2-a.svg" is not declared in the OPF manifest.
ERROR(RSC-008): aata.epub/EPUB/xhtml/matrix-section-groups.xhtml(84,882785): Referenced resource "EPUB/xhtml/generated/latex-image/matrix-r2-translations-a.svg" is not declared in the OPF manifest.
ERROR(RSC-008): aata.epub/EPUB/xhtml/matrix-section-symmetry.xhtml(84,49422): Referenced resource "EPUB/xhtml/generated/latex-image/matrix-glide-reflections-a.svg" is not declared in the OPF manifest.
ERROR(RSC-008): aata.epub/EPUB/xhtml/matrix-section-symmetry.xhtml(84,575523): Referenced resource "EPUB/xhtml/generated/latex-image/matrix-p4m-p4g-a.svg" is not declared in the OPF manifest.
ERROR(RSC-008): aata.epub/EPUB/xhtml/actions-section-burnsides-counting-theorem.xhtml(84,587625): Referenced resource "EPUB/xhtml/generated/latex-image/actions-switching-two-variables-a.svg" is not declared in the OPF manifest.
ERROR(RSC-008): aata.epub/EPUB/xhtml/boolean-section-algebra-of-circuits.xhtml(84,57874): Referenced resource "EPUB/xhtml/generated/latex-image/boolean-one-zero-a.svg" is not declared in the OPF manifest.
ERROR(RSC-008): aata.epub/EPUB/xhtml/galois-section-fund-theorem-galois-theory.xhtml(84,540727): Referenced resource "EPUB/xhtml/generated/latex-image/galois-root3-root5-a.svg" is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/actions-switching-two-variables-a.svg" exists in the EPUB, but is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/boolean-one-zero-a.svg" exists in the EPUB, but is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/galois-root3-root5-a.svg" exists in the EPUB, but is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/matrix-glide-reflections-a.svg" exists in the EPUB, but is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/matrix-o2-a.svg" exists in the EPUB, but is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/matrix-p4m-p4g-a.svg" exists in the EPUB, but is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/matrix-r2-translations-a.svg" exists in the EPUB, but is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/matrix-sl2r-a.svg" exists in the EPUB, but is not declared in the OPF manifest.
WARNING(OPF-003): aata.epub(-1,-1): Item "EPUB/xhtml/generated/latex-image/permute-transpositions-cube-left.svg" exists in the EPUB, but is not declared in the OPF manifest.

In theorem 17.17, the polynomial is not assumed to be monic. It then refers to Gauss's lemma (theorem 17.14), but that one assumes the polynomial is monic. Only in the next chapter is a more general version of Gauss's lemma proved.

Chapter 6, Exercise 13
In line 155 of source /aata/src/exercises/cosets.xml the exercise references Theorem~6.3.
This needs to be changed to an <xref>, though I am unsure that 6.3 is the correct number as currently 6.3 is a Lemma.

Chapter 19, Exercise
In line 346 of source /aata/src/exercises/boolean.xml the exercise reference Theorem~19.14.
This needs to be changed to an <xref>, though I am unsure that 19.14 is the correct number as the exercise is referencing Boolean algebras, however 19.14 is about lattices.

From other sources I gather that the expression given is only a cyclotomic polynomial when n is prime, while the exercise puts no restrictions on n.