2016 November 06

originally posted on facebook

Floridian friends-

In 2000, at least 991 Nader supporters in Florida pledged to vote for Al Gore in exchange for Gore supporters in other states voting for Nader. These vote swaps very nearly decided the election: the final difference between Bush and Gore was 537 votes.

The 2000 vote swaps were organized through a handful of websites that were rapidly shutdown under legal threats. Since then, it has been determined in court that vote swapping is protected under the first amendment, and vote swapping has re-emerged as a way for third-party supporters in swing states to make their vote count for even more, and for main-party supporters in non-swing states make their vote count for /something/.

If you or someone you know are planning on voting for a third-party candidate in a swing state, you can greatly increase the power of your vote by vote swapping. All you have to do is contact a Clinton supporter you know in another state and suggest a trade. Or a variety of resources exist (see below) for finding voting partners online; these sites generally are limited by needing more swing-state voters, and some have offered 2-for-1 trades for people from swing states, so you can turn your third party vote into two third party votes.

General overview:

Wikipedia

Major vote swapping sites:

https://voteswap.us/ (prefers matching you with facebook friends)

https://trumptraders.org/

https://www.makeminecount.org/

https://apps.trimian.com/nevertrump

Discussion of vote swapping:

http://www.scottaaronson.com/blog/?p=2891

Vox

2016 October 30

originally posted on facebook

The best classical music quiz!

https://www.sporcle.com/games/iglew/cant-read-music

2016 October 20

originally posted on facebook

I’ve seen many people state that they find the flaws in both major party candidates so troubling that they cannot in good conscience vote for either one. They have a point. That’s why I, as a single issue voter, am voting for the only candidate who I can truly support on the issue that matters to me: Lincoln Chafee, champion of the metric system! With 21% of likely voters supporting the metric system, I have every confidence in our success.

2016 October 18

originally posted on facebook

Today I read an economics paper: “Japan’s Phillips Curve Looks Like Japan”.

http://qed.econ.queensu.ca/working_papers/papers/qed_wp_1083.pdf

2016 October 07

originally posted on facebook

Leaked video of former US Senator Gordon Smith of Oregon talking with Mormon leaders (the Mormon “apostles”) reveals details I find disturbing:

This is part of a larger series of leaked videos of Mormon leaders, another of which features the only time I’ve heard “homosexual agenda” used seriously.

https://www.wweek.com/news/2016/10/03/mormon-leaks-video-features-former-u-s-sen-gordon-smith-r-ore/

https://www.reddit.com/r/exmormon/comments/55iq5z/the_gordon_smith_leak_is_scandalous/

http://archive.sltrib.com/article.php?id=4424105&itype=CMSID

*In several places the term “temple recommend” is used. This is approval to receive the endowment [1] ceremonies, and indicates a Mormon member in good standing. In the endowment, members swear [2] to dedicate everything, including their lives, to the church. Until 1990 the ceremony included specific oaths [3] to sooner kill oneself than reveal details of the ceremony. Until the 1930s the ceremony included a specific oath [4] to take vengeance on the US for the death of Joseph Smith. Mormons believe that only those who have undergone endowment are admitted to (the highest circle of) heaven.

[1]  Wikipedia

[2]  Wikipedia

[3]  Wikipedia

[4]  Wikipedia

2016 August 18

originally posted on facebook

My favorite fact in elementary number theory is that odd primes can be written as a sum of two squares if and only if they are congruent to 1 modulo 4. This has been known since 1634 and proven (by Euler) in 1747. However it is a very difficult fact of number theory that asymptotically half of all primes are congruent to 1 mod 4 (the other half being 3 mod 4); this is a consequence of Chebotarev’s density theorem [1], proven in 1922. Combining we find that asymptotically half of all primes are a sum of two squares.

What about numbers in general? It turns out that the asymptotic density of numbers which are a sum of two squares is 0. This follows from a theorem of Landau and Ramanujan in 1906 [2]. The proof appears not to be available online but there are some simple arguments that suggest the asymptotic density is 0.

So why the difference? Why are numbers of the form a^2 + b^2 more likely to be prime?

[1]  Wikipedia (and see also Wikipedia for fun)

[2]  Wikipedia

2016 August 14

originally posted on facebook

I ran into a very interesting (and fairly readable) paper that demonstrates remarkable algorithms for (1) finding the maximum of a continuous function on the real interval [0, 1] and (2) integrating a continuous function over the interval [0, 1]. What is interesting about this is that these algorithms work with exact reals, which represent real numbers as lazy bit streams (so the algorithms take a finite number of steps to compute the first k bits of the answer). Underlying these algorithms is “Berger’s search operator” which computes in finite time whether a proposition is true for all real numbers in [0, 1]. By itself, that is already remarkable! This function is just 5 lines of code (see Figure 1).

https://www.research.ed.ac.uk/portal/files/12417914/download_2.pdf

The author writes “The integration algorithm performs abysmally on any interesting functions. The maximum algorithm performs a little better.”. I would assume the speed problem is due to the Berger search being hopelessly slow (as it has an exponential blow-up in complexity as more bits become relevant). I have found another paper that gives substantially faster search algorithms for this problem, but unlike the previous paper, is not at all readable (at least to me). An interesting remark from the conclusion of the second paper: “this work is a genuine application of topology to computation: theorems formulated in the language of computation, proofs developed in the language of topology.”.

http://www.cs.bham.ac.uk/~mhe/papers/exhaustive.pdf

2016 August 08

originally posted on facebook

Here’s a cool (and very simple) data structure I just learned about that takes advantage of uninitialized memory to use less time than space!

https://research.swtch.com/sparse

Maximally symmetric 4-colorings of a pentagonal hexacontahedron

2016 July 05

originally posted on facebook

(This was prompted by a friend asking for symmetric ways to 4-color a pentagonal hexacontahedron.)

https://math.stackexchange.com/questions/1850376/coloring-the-pentagonal-hexecontahedron

Let S be the group of icosahedral rotational symmetries. Write PH for pentagonal hexecontahedron.

  1. If you are looking for nontrivial symmetries in S that preserve a coloring of a PH, there are none, as any rotation swaps some two adjacent faces which would then have to have the same color.

  2. If you are looking for nontrivial symmetries in S that permute a 4-coloring of a PH, you can probably find them with trial and error… you don’t have to consider 5-order symmetries as 5 is prime and larger than 4.

  3. More interesting is what I think you are looking for: a 4-coloring of a PH whose group G of symmetries that permute its colors are transitive (that is, for any pair of colors there is a symmetry in S that swaps those two colors). So G is a subgroup of S: we can see the subgroups of S here: Wikipedia But G is also a group of permutations of four colors, so can be viewed as a subgroup of S4 https://en.wikiversity.org/wiki/Symmetric_group_S4 This is our leverage to getting information about G.

Now since G is a subgroup of both S (with 60 elements) and S4 (with 24 elements), its order must be a divisor of gcd(24, 60) = 12. Call the four colors A, B, C, D. Since for each color k there is an element of G that sends A to k, there must be at least four elements of G, so G is order 4, 6, or 12. Looking at the subgroups of S4 with order 6 we see that none of them are transitive (in each case there is a color that is not swapped to any other color by any element of G), so G has order 4 or 12.

There are 7 subgroups of S4 of order 4. Three of them are isomorphic to Z4, and Z4 does not appear as a subgroup of S (S has no 4-rotations) so we only need to consider the four isomorphic to V (the Klein four group). The first three are identical (up to internal automorphisms of S4) so really we are only considering two different cases where G has order 4.

In the first case, G is isomorphic to V and contains a transposition (that is, an element that swaps some two colors A and B while preserving colors C and D). PH has 60 faces, and any 2-rotation groups those 60 faces into 30 pairs which are swapped under rotation. For a rotation to preserve a color under swapping, that color must appear on an even number of faces. But A, B, C, and D all appear on the same number of faces (as they are permuted transitively), which must be 60 / 4 = 15. This is impossible, so G contains no transpositions.

We are down to two cases: either G has order 12, in which case it is A4 (contains every even permutation of the four colors), or it has order 4 and is isomorphic to V and contains the even permutations of order 2 (that is, the elements of G are the identity; swapping A <-> B, C <-> D, swapping A <-> C, B <-> D, and swapping A <-> D, B <-> C). Note that the former group contains the latter group, so if we fail to find the latter group at all then we know neither is possible.

I went ahead and looked for 4-colorings that have their colors permuted in 12 different ways (i.e. have A4 or tetrahedal symmetry), which is the most possible. In fact, if I haven’t made any mistakes, there are exactly 4 such colorings! So there do exist highly symmetric 4-colorings of the PH.

I’ll try and give a written procedure. First G = A4 appears 4 different ways inside of S so you need to choose which one you will use for the symmetries of your coloring. (There are four ways you can put a tetrahedron inside of a PH so that the symmetries of the tetrahedron are also symmetries of the PH.) It will have 3 axes of 2-rotation, all perpendicular to each other, and 4 axes of 3-rotation, cutting diagonally with respect to the 2-rotation axes. We will ignore all other rotation axes during this discussion, we are only interested in rotation axes for elements of G.

Each axis of 3-rotation preserves one of the colors and cycles the other four; you can choose which does which color arbitrarily (so long as each 3-axis preserves a different color). Keep track of which color each 3-axis preserves as we will need it momentarily.

First we color every face which is touching a 2-axis as follows. For each such face F1, find the face F3 that is touching a 3-axis that shares a vertex but not an edge with F1. Give F1 the color preserved by that 3-axis.

Now we color each set of 5 faces which share a vertex. Going around clockwise call the faces F1, F2, F3, F4, F5 where F1 touches a 2-axis and F3 and F5 touch a 3-axis. (If this is not possible then your PH is flipped relative to mine so use anticlockwise.) Let A be the color fixed by the axis touching F3. Let C be the color fixed by the axis touching F5. Let B the color of the face sharing an edge with F1 that is already colored in the last step. (It is the color that the axis touching F1 sends A to.) Let D be the other color. Then, depending on which of the four colorings you are doing, the colors of F1 through F5 are:

Choice 1. A-D-B-A-D

Choice 2. A-C-D-A-D

Choice 3. A-C-D-C-D

Choice 4. A-C-D-C-B

In total, four choices of which copy of A4 to use, 24 choices of which color each axis preserves, and 4 nonequivalent colorings, makes 4 x 24 x 4 = 384 different ways of putting colors on ignoring symmetries.

2016 June 29

originally posted on facebook

Some adjectives used to describe certain types of cardinal numbers, according to Wikipedia:

small; large; worldly; inaccessible; hyper inaccessible; reflecting; totally indescribable; unfoldable; shrewd; ethereal; subtle; almost ineffable; ineffable; totally ineffable; remarkable; Ramsey; ineffably Ramsey; completely Ramsey; strongly Ramsey; super Ramsey; measurable; strong; tall; superstrong; extendible; almost huge; super almost huge; huge; superhuge

Wikipedia

2016 June 24

originally posted on facebook

For every prime p > 10^{75}, there exists an infinite group whose nontrivial subgroups each have p elements. There is no such group for p = 2 or p = 3.

Wikipedia

2016 April 08

originally posted on facebook

This morning I wrote a Newton fractal generator:

http://imgur.com/a/phf25

http://gfycat.com/MistyFoolishHawk

Wikipedia

2016 April 07

originally posted on facebook

A quick exercise in L’Hopital’s rule: compute the derivative of

W((x^2 + x - 1) e^(x - 1))

at x = 0, where W is the Lambert W function. The derivative of W is given by:

\frac {dW}{dz} = \frac {W(z)} {z (1 + W(z))}

Wikipedia

Wikipedia

(If you wish to be technical about differentiating a function at its branch point, you can consider only x > 0 and say that you are taking the limit of the derivative as x goes to zero. Or you can just ignore this note.)

2016 March 07

originally posted on facebook

Today I fixed a bug in my lua code by adding the lines:

if x == 0 then
    x = 0
end
if y == 0 then
    y = 0
end

Turns out lua has a “negative zero”, which is equal to zero and behaves almost the same except when printed to a string. I used such strings as keys in a hash map which then caused the program to hang in an infinite loop because its logic about what was in the hash map was wrong.

2016 March 04

originally posted on facebook

I have on occasion recommended the excellent book Governing the Commons by Elinor Ostrom (Nobel prize winner in economics, and the only woman to have done so), most especially to those interested in the management of natural resources and the role of government regulation. The book is very accessible and presents many real-world examples of successful and failed management, giving a more nuanced view of the so-called “tragedy” of the commons.

Now, (the essence of) this book is available in an even more accessible format: a six-minute youtube video!

YouTube

The succeeding video discusses the specific case of the decline of Atlantic cod off the Grand Banks.

YouTube

These videos, like the others on the channel, are well-researched and thought out and an excellent model of how lay-science should be. Watch!

2016 February 22

originally posted on facebook

A handful of you may remember a lecture I taught on the J programming language way back at Caltech. Here is an interesting 1975 tutorial on APL, “A Programming Language”, which was the predecessor of J.

YouTube

Conway’s Game of Life in one line of APL:

YouTube

http://patriciopalladino.com/blog/2012/08/09/non-alphanumeric-javascript.html

A poem based on Dwarf Fortress bug reports:

https://medium.com/@henryoz/issues-a-dwarf-fortress-found-poem-4a6c3cb33232

2016 February 03

originally posted on facebook

I made a computer in a computer game. Wanted to do this with Dwarf Fortress a while ago but never got far (and that’s been done several times already), maybe I’ll return to that someday.

https://www.reddit.com/r/factorio/comments/43giwy/i_made_a_programmable_turingcomplete_computer_in/

https://imgur.com/a/MU3T9

2016 January 21

originally posted on facebook

A new Mersenne prime and a new planet? What a week!

“it was sent to the four most renowned Australian number theorists. They were asked to work on it for six hours. None of them could solve it in this time. […] the jury finally had the courage to choose it as the last problem of the competition. Eleven students gave perfect solutions.” Wikipedia

“Discombobulate” was originally “discombobricate”.

https://en.wiktionary.org/wiki/Appendix:Fanciful_19th_century_American_coinages

Finding integer solutions to

\frac a{b + c} + \frac b{c + a} + \frac c{a + b} = 4

https://mathoverflow.net/questions/227713/estimating-the-size-of-solutions-of-a-diophantine-equation

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