venhance / napkin

An Infinitely Large Napkin

Napkin v1.5

These are the source files for Evan Chen's An Infinitely Large Napkin. You can read about the project on my webpage.

Download

You can download the most recent PDF or generated log files. (Thanks to aDotInTheVoid for helping me set up the Travis CI to automate this process!)

Code

The project can be compiled on a system supporting latexmk and asymptote, with a sufficiently recent version of TeX Live. You'll need to mkdir asy, then simply run latexmk. On a system with nix, just run nix build. (Or, see .travis.yml for the instructions that Travis CI uses to compile.)

Pull requests are welcome! You can email corrections directly to me as well.

Coffee

If you like this project and want to support me, you can consider buying me a coffee.

napkin's Issues

Completion of the Probability Unit

This is just a tracking issue.

21E hint is defective

Amol (orz) and I have agreed that the hint doesn't actually make much sense. This can be seen as this implies by taking $g = h$ that all $C_g$ have the same conjugacy class size, which is evidently false. It's likely best to replace this with the standard solution that $AB = D \iff BA = D$ for square matrices $A, B$ and diagonal matrix $D$.

Vector instead of scalar

napkin/tex/linalg/inner-form.tex

Line 301 in 408e4de

Then \[ 0_V = \left< v_1, \sum a_i v_i \right> = \ol{a_1} \norm{v_1}^2. \]

Should be 0 instead of 0_V.

The hint of v1.5's problem 51C "might" be wrong

NOTE: This issue was actually found by @puzzlet

Problem 51C is:

And hint of problem 51C is:

napkin/tex/alg-NT/galois.tex

Lines 614 to 617 in 1a3fc0a

	\begin{hint}
	Left-hand side has minimal polynomial of degree $7$,
	but the right-hand side lives in a degree six extension.
	\end{hint}

The description that the left side has a minimal polynomial of degree 7 is presumed to mean the polynomial below.

But I'm afraid it is not minimal polynomial since it is not irreducible.

How can I easily solve 51C with Galois theory?

Can someone define $f$ in this context?

napkin/tex/linalg/dual-trace.tex

Line 457 in 1114f56

by ``collapsing'' each pure tensor: $f \otimes v \mapsto f(v)$.

Add AIME 2018 I 10 after Problem 20C

Mainly a reminder for myself to do this as a PR. See the third post in https://artofproblemsolving.com/community/c5h1604152 for context.

Limit chapter typos

Silly example of monotonicity has typos: spelling of monotonicitiy as well as the sequence provided is not monotonic.
https://github.com/vEnhance/napkin/blob/master/tex/calculus/limits.tex

Partial sums of nonnegatives bounded implies convergent
... sequence s_n converges to M=sup_n s_n
but M is just an upper bound for the sequence.

Translation - l10n

Hey there!
I've been working on napkin for a while, and I suppose to translate napkin to languages like Chinese so more people can use this great resource. With the same open-source license, I can work on translating this into Chinese. I'll pull a new request (a new branch or simply a l10n folder) to the project soon. :D

Randomised hint order

I really like how you randomised the hints in EGMO. I feel like it would be a good idea to apply this to Napkin too. Is there a reason this wasn't done?

Cycle decomposition notation

The solution sketch of problem 1C is using what I now know to be the cycle notation for permutations. However, I stumbled upon this by accident - I haven't encountered this notation beforehand, which made the solution sketch really confusing. I also haven't been able to find the notation in the notation glossary.

Weird Sentence in Chapter 20 intro

should this be "they exhibit the good property then we just have to figure out ..." instead of that? The whole sentence feels pretty clunky to me.

HTML version?

First of all, thank you for this amazing book(?). I wish I had known about napkin before my calculus classes.

I was wondering if there was a possibility for an HTML version for the same.
This would help in 2 ways -

• Permalink-ing to chapters/sections/theorems/... via links.
• Directly hosting the html copy on github.io for speedy reference.(like a gitbook/readthedocs)

This could've been as easy as htlatex Napkin.tex but I was not able to build it successfully.
The Travis CI uses some specific versions for everything (Ubuntu bionic, texlive ppa) and I was not able to build the same on my config.

(I expect everything but the diagrams to work well in HTML. Will try to reproduce the build environment on a Virtual Machine tomorrow, hope it succeeds.)

Proof of Second Orthogonality

The following problem features the second orthogonality relation:

Many books and notes[1][2] proves it by treating character table as a square matrix.

The Napkin's approach in the Hint section seems a bit related but I don't see how:

Is this equivalent to other proofs? If so, how?

Inconsistent notation: |G/H| and [G:H]

For G and H groups, I notice two different notations being used:

• sometimes |G/H| is used:

• some other times, [G:H] is used:

(the latter notation is also used for the degree of a field extension however.)

Is there some reason behind this, or should it be cleaned up (in the latter case which one do you prefer?)

Answer to 5F seems to have a counterexample

I may be wrong about something, as I'm pretty lapsed in math and keeping up with this book the best I can.

I was surprised by the answer to the second part of 5F, the prime ideals of the real numbers, which seemed to kind of come out of nowhere without explanation. I tried to look up more about it, particularly the case of (x^2 - ax + b) when that quadratic has "two conjugate real roots".

Was this supposed to say "two conjugate non-real roots"?

I found this topic on Math Overflow, which says that (x^2 + 1) is prime, because ℝ[x]/(x^2 + 1) is ℂ. The roots of (x^2 + 1) are ±i, which are not real.

Theorem 19.5.7 bug?

This is about Schur's lemma for algebraically closed fields. As stated it claims if V is an irrrep, then any intertwining from V to V must be a scalar multiplication. However this seems to only be true if the intertwining is from V to V where we're considering the same representation in the domain and image. The "same representation" condition is not mentioned.

https://github.com/vEnhance/napkin/blob/90cd09947af7ec53138a88b44a31cf4cc40ddd8a/tex/rep-theory/rep-alg.tex#LL470C1-L476C14

Gorin + Tang notes offline; fix links

Caught by houlsimp

Tiny typo: \RR not \QQ

napkin/tex/calculus/integrate.tex

Line 131 in 0c80de3

to a function $\wt f \colon \QQ \to N$.

The referenced line should be \RR not \QQ.

Minor: typo on page 721 in latest version

In Definition 75.4.1, the second part "Gluing", the sections s_α should be in F(U_α) instead of in (U_α), because s_α are the elements of the local sections.

Exercise about ideals before the definition of ideals

In Exercise 4.4.5, you ask the reader to prove that all fields have two ideals. However, this is before ideals are defined, and as a consequence, one cannot possibly know how to solve this yet.

Consider adding a license and/or CLA

Hi, I've been enjoying going through your book while sheltering in place. Thanks so much for the work you've put into it.

I noticed there isn't an explicit license for this book (that I can find). IANAL but I've worked a lot in open-source software, including on licensing, so take this unsolicited advice (that you may already be aware of) for what it's worth.

In the absence of an agreement otherwise, when I send you a PR, I retain copyright over my contribution. If the contribution is "meaningful", then I can assert rights over that contribution, like preventing you from publishing a book which contains it, or like demanding that you stop distributing a work which contains that contribution online.

A contributor license agreement (CLA) would basically say, in order to send me a PR, you need to give me (Evan) the rights to your work. So I can do whatever I want with it in the future. (I guess, you could have a CLA which says something else, but that's what most CLAs say.)

Alternatively or in addition, you could add a license to the work. If you don't have a CLA, this would say, I (Evan) agree to let anyone use this work under X terms, and by contributing, you retain copyright, and you also agree to let anyone use this work under the same terms.

If you had a license plus a CLA, this would mean, I agree to let anyone use the current version of this work under X terms. I can't revoke the terms (that's part of most licenses anyway), but I might decide to change the terms for future versions, or add a new license that you could choose to use the old versions under. Those are things I can do because I own the copyright for the whole thing.

Making either of these changes is unfortunately work, because you'd have to go back in time and get everyone who has contributed to the book to agree to the license and/or CLA. But they're also easier to do earlier than later.

One other wrinkle: I notice that there are two GPL'ed tex packages in the repository. This is now slightly more speculation on my part, but I suspect that this means that the whole book is covered under the GPL. That has all sorts of fun consequences.

Anyway, I don't mean to punish a good deed. Hope you're staying safe and healthy, and thanks again for your book.

Make use of git tags and github releases

The readme says "Napkin v1.5", but the tags and releases sections are empty.

It would be nice to do actual releases :)

Problem 4H requires Modules which appear later in the document

It's probably possible to simply move this problem to being in the PID Structure Theorem section.

Reps of k[x, y] by V require representing operators to commute

In the example about algebra representations, the following is written:

napkin/tex/rep-theory/rep-alg.tex

Lines 208 to 211 in 343bd11

	\ii We can create arbitrary ``functional equations'' with this pattern.
	For example, if $A = k[x,y] / (x^2 - x+y, y^4)$
	then representing $A$ by $V$ amounts to finding operators
	$S, T \in \Mat(V)$ satisfying $S^2 = S-T$ and $T^4 = 0$.

However to my knowledge, since the polynomial ring k[x, y] is commutative, the representing operators S, T must commute as well. This should be mentioned.

diffgeo change of variables

Explain how the change of variables nonsense in 18.02 follows beautifully from
the morally correct and elegant setup in this textbook 😇 /s

Adding hyperlinks to chapters in advised reading graph.

In page 9 of the pdf or ix, Advice for the reader, Deciding what to read there is a graph of what topics and chapters are prerequisites to other topics and chapters. I tried to click to go to the Calc topic and noticed that it was just text. I suggest that it links to the first page in the range of chapters in the topic.

The relevant code should be tex/frontmatter/digraph.tex.

Possible typo on page 573

It is written that covering projection p is injective, but shouldn't it be that p_* is injective (p is not injective since in your own example 59.1.4 you demonstrate it)?

Build instructions

Thanks for making this available!

I think it would be useful to provide build instructions (via a Makefile?) and list non-typical CTAN package dependencies (e.g., prerex, diagrams [?], ...).

Rasiowa-Sikorski lemma

The original Rasiowa-Sikorski lemma demands the family of dense subsets to be countable. It seems that Napkin also assumes it ("hit them one by one") but are we sure that the family of all dense sets of P is countable if the model M is countable?

section headings in contents overlap enumeration

split single pdf into several pdf files

Now napkin is almost 1k pages, do you have a plan to split this pdf file into several pdf files,
like Milne note It is easy to read.

Add a bit of hint for Example 25.3.2 on where "2" comes from

Hi thanks for the wonderful book! When reading 25.3.2 it takes me a bit of time before finally understanding U_x. Maybe "build a circuit U_x ... to |k> |2^k mod M>" can be rewritten as "build a circuit U_x ... to |k> |x^k mod M>". Otherwise, firstly I thought the "2" is just because we are dealing with binary numbers.

I think 12.2 (definition of determinant) omits an important part of the proof

12.2 defines the determinant by defining a function $\wedge^n(V) \to \wedge^n(V)$ by taking $v_1 \wedge v_2 \wedge \ldots \wedge v_n$ to $T(v_1) \wedge T(v_2) \wedge \ldots \wedge T(v_n)$. This is not too hard to show as linear, but it is actually pretty hard to show that this is even defined as a function, given $(cv) \wedge w = c \wedge (cw)$ and so on. For a general wedge space, T has to be (at least almost) linear for this to map equivalent elements to equivalent elements, and I think a lot of the reason that matrices have determinants at all is hidden in this step of the proof. For it to work, if two parallelograms have equal orientation and area (and are the same in the wedge product space), applying T must map each of them to the same new orientation and area. I didn't really make as much sense to me the first time I read the section, because we're just suddenly on "linear maps are linear in wedge spaces" without much buildup and I couldn't see how we got there.

Change visual style of hyperlinks

Wouldn't it be nice to change the appearance of hyperlinks to something less intrusive?
Maybe use colorlinks instead?
Like shown here: https://tex.stackexchange.com/questions/823/remove-ugly-borders-around-clickable-cross-references-and-hyperlinks

File `media/tangent.pdf' not found.

It apears you have accidentaly .gitignored one of your images.

Invocation: pdflatex -halt-on-error -interaction=nonstopmode Napkin

LaTeX Warning: File `media/tangent.pdf' not found on input line 73.


! Package pdftex.def Error: File `media/tangent.pdf' not found.

See the pdftex.def package documentation for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.73 	\includegraphics{media/tangent.pdf}
                                         
!  ==> Fatal error occurred, no output PDF file produced!
Transcript written on Napkin.log.

Full log

Unmentioned caveat: G/H × H is not necessarily isomorphic to G

When you introduce quotient groups, you show some examples of how × (the direct product) and / behave somewhat like multiplication and division: (G × H) / G' ≅ H.

I believe it is quite important to also show the non-examples. As I've realized when I learned of the Schur-Zassenhaus theorem, it is not necessarily the case that (G/H) × H ≅ G (Wikipedia has a nice counter-example here, take G = Z/4Z = {0, 1, 2 3} and H = <2>).

Minor: typo on page 260

"We could have used other bases, like |→⟩A ⊗ |0⟩B and |←⟩A ⊗ |0⟩B for the first eigenpsace, but it doesn’t matter."

When will the volume of "Algebraic Geometry III: Schemes" be written?

Hi thanks for the book, I wonder when will "Algebraic Geometry III: Schemes" be written - it looks quite interesting!

configure github sponsors

What a phenomenal project and artifact!

Please consider setting up github sponsors so that you can receive funds from any github user who already has payment information on file. Github takes no cut.

Once configured there will be a sponsor button in the upper right , you can see it in action on the Intellij Elixir Plugin page.

And thanks for using the GPL so that it can be extended and remixed.

Cant find appendix

Hi, this is probably a very dumb question, but I can't seem to find the appendix for the book.

I've searched through the book contents, through this code and looked online to no avail. An appendix is mentioned, but doesn't seem to be provided.

Is there a Napkin-like book for physics?

Hi thank you for the book, which is quite helpful and I enjoy reading it! I wonder whether there exists a Napkin-like book for physics in the world?

Minor error constructing outer measures from pre-measures

https://github.com/vEnhance/napkin/blob/main/tex/measure/caratheodory.tex#L215-L217

The construction linked above isn't quite sufficient to force the $B_n$ to be disjoint. Instead, the line $B_3 = (A \cap A_3) \setminus B_2$ should read $B_3 = (A \cap A_3) \setminus (B_2 \cup B_1)$.

Give some motivation for the definition of algebraic integer?

While the definition of algebraic number is quite obvious, I feel the definition of algebraic integer is a bit unmotivated.

I think something like the following.

https://math.stackexchange.com/questions/693541/intuition-behind-the-definition-of-algebraic-integers

---

other points:

• we may define that if K = ℚ[α], then 𝒪_K = ℤ[α] for α the "simplest" representation in some sense, but this definition depends on the choice of α and doesn't work out -- for example, if α is √d or a root of unity, it's natural to choose 𝒪_K = ℤ[√d] with squarefree d, but what if α is something like the root of X³-X²-2X-8?

  (remark: this appear in the counterexample of monogenic extension)

  In other words, we want the definition of algebraic integer to be some intrinsic concept of the numbers.

• the algebraic integers should be, in some sense, "sparse" in K, just like how ℤ[i] is sparse in ℚ[i].

  However, what "sparse" means here is not very clear (unlike the case ℚ[i] → ℂ ≅ ℝ² has a natural embedding -- for example, in the case of K = ℚ[√2], K itself is dense in ℝ), so we require the following:

  If we pick any ℚ-vector space basis {β₁, …, βₙ} of K, then there is some ε > 0, such that for all elements c₁ β₁ + ⋯ + c_n β_n ∈ 𝒪_K, then c₁²+c₂²+⋯+cₙ² > ε².

The last point is the important point here, and it turns out to be equivalent to each of the 3 of the following:

• for all α ∈ 𝒪_K, ℤ[α] is a finite ℤ-module.
• minimal polynomial of α has integer coefficients.

---

(remark: another possible definition of "sparse" you can think of is that (𝒪_K ∩ ℚ) is not dense, equivalently, 𝒪_K ∩ ℚ = ℤ. While this is a corollary of the above, the converse need not hold, for example ℤ[(i√7-1)/4] ∩ ℚ = ℤ, but (i√7-1)/4 is not an algebraic integer)

---

Another point, it may be useful to mention that this definition of algebraic integer is useful in the sense that anything smaller than it does not have unique factorization of ideal property. (https://math.stackexchange.com/questions/682912/ideals-in-a-non-dedekind-domain-that-cannot-be-factored-into-product-of-primes/682947#682947) i.e. proving unique factorization of ideal implies integrally closed, if the proof is nice.

Are we missing a letter? (can make a PR)

By the matrix multiplication I know, it seems that the second formula lacks coefficient x_k.

https://github.com/vEnhance/napkin/blob/main/tex/quantum/shor.tex#L92-L96

Compilation Error: Cannot determine size of graphic

The exact error is:

! LaTeX Error: Cannot determine size of graphic in media/kofi4.png (no Bounding Box).

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                              
l.18 ...ludegraphics[width=32ex]{media/kofi4.png}}

I think this isn't a big deal based on this TeX.SE post. I wanted to ask here if it's even worth fixing. Like, I think I'm only experiencing this because I'm using TeXLive, and so using latexmk compiles to DVI by default. Compiling with latexmk -pdf works fine without errors. Or specifying $pdf_mode=1 in the latexmkrc file works too.

Reference broken (missing) and Possible Typos

I have noticed something when reading it:

1. For problem 1A., the figure's source [Ge] "Topological Girl’s Generation. Topological Girl’s Generation. url: https: //topologicalgirlsgeneration.tumblr.com/." is dead.
2. In Example 4.4.4 (b), the phrase is weird: it should be "which we will usually denote" or something like that.

Dubious formatting in Remark 9.4.9

Remark 9.4.9 looks like this:

The combination of an em-dash and bullet on the same line strikes me as odd, and I suspect it happened by accident.

Too much just

I love this book, but the just count could be culled a bit

Typo: should have >= instead of =

Hi,
On page 208 in the newest version in the proof for the proposition, I think it should be "n_r \geq pq" at the first sentence of the second paragraph instead of "n_r = pq", especially since you say a bit later that "we must have ... at least pq Sylow r-subgroups". This confused me when I was reading the proof. Correct me if I'm wrong.

Theorem 29.4.5 is broken

See e.g. https://mathoverflow.net/questions/81613/non-analytic-function-with-convergent-taylor-series-everywhere

I forget what I meant to write