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Dan Christensen
9,809 followers -
Interested in math, physics, computation and more.
Interested in math, physics, computation and more.

9,809 followers
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Interesting story of an Osprey's journey. I hope we learn whether she returns to Detroit in 2019!
In 2016, a young osprey named Julie was banded in Michigan with a small GPS beacon on her back. Follow her amazing, unlikely migration journey in this incredible story map. http://p.ctx.ly/r/7l4g #WorldMigratoryBirdDay
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The recent conversations between Google's assistant and businesses are really amazing. Try clicking the first two audio links in the pages I've linked to. They are less than a minute each and are very impressive.

https://ai.googleblog.com/2018/05/duplex-ai-system-for-natural-conversation.html
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+Chris Kapulkin and I are organizing an online seminar series, the Homotopy Type Theory Electronic Seminar Talks. The series is aimed at people who already have background knowledge in homotopy type theory. The next speaker is Emily Riehl, whose title is "The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories" (yes, that's correct). The talk is at 11:30 Eastern time on Thursday, March 1. More details at the seminar web page:

http://uwo.ca/math/faculty/kapulkin/seminars/hottest.html
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The first of the Homotopy Type Theory Electronic Seminar Talks, with Peter Lumsdaine, went very well, and the video is now available on YouTube.
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I'll be at the AMS meeting in San Diego from Thursday morning until Sunday morning. If you'll be there and want to meet up, let me know!
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My son worked with Everbloom Studios to create a Christmas themed game in Minecraft. Today (Boxing Day) Ultimate Pig Race is the featured game in the Minecraft Store!

Here's the blurb:

Santa needs help from his magical pigs to deliver presents!
Race against the clock or play with up to 9 of your friends! 12 unique and
challenging tracks to play (6 singleplayer, 6 multiplayer), three types of
adorable pigs, and awesome powerups!

http://everbloomstudios.com/maps/ultimate-pig-race/
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Reading one of +Greg Egan​'s novels is always one of the highlights of my summer. I'm looking forward to this one!
Dichronauts

My new novel Dichronauts is officially published in the US on July 11, but copies are already in stock at Amazon, and it might already be available from your favourite bookshop.

You can read an excerpt by following the link below, and supplementary material about the universe of the novel by following the links from that page.

http://www.gregegan.net/DICHRONAUTS/E1/Excerpt.html
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Game theory and partial orders

This post announces a paper that disproves a longstanding conjecture in game theory and also computes a 138 digit number related to partially ordered sets.

Let n be a natural number, and consider the following two-player game, called Subset Takeaway. The players take turns naming a non-empty proper subset of {1, ..., n} that doesn't contain any set named earlier. If a player has no legal move, they lose. For example, when n = 2, the second player will always win, since if the first player names {1}, the second player will name {2}, and vice versa. For n = 3, the second player can win by naming the complement of what the first player names, leading to a position equivalent to the n = 2 starting position. In 1981, Gale and Neyman showed that the same is true for n = 4 and n = 5, and conjectured that this pattern continues. In 1995, Mark Tilford and I showed it is true for n = 6, using a computer.

Andries Brouwer and I showed that for n = 7, the conjecture is false. This was checked by two independently written programs.

In addition, we realized that our techniques were exactly what is needed to make the following computation. Write P(n) for the set of all subsets of the set {1, ..., n}, a partially ordered set under inclusion. Count the number of total orders on P(n) which are compatible with the partial order. The answer was known for n ≤ 6, and we were able to compute it for n = 7, resulting in the 138 digit number shown in the last line below.

0 1
1 1
2 2
3 48
4 1680384
5 14807804035657359360
6 141377911697227887117195970316200795630205476957716480
7 630470261306055898099742878692134361829979979674711225065761605059425237453564989302659882866111738567871048772795838071474370002961694720

All of this is described in

Counterexamples to conjectures about Subset Takeaway and counting linear extensions of a Boolean lattice
Andries E. Brouwer, J. Daniel Christensen
https://arxiv.org/abs/1702.03018

#arXiv arXiv:1702.03018

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+Dan Piponi wrote an entertaining description of the functional exponential and logarithm of power series under composition, and gave concise Haskell implementations.

I'm reminded of Pierre Cartier's Mathemagics paper, which starts with exponentials and leads to the time ordered exponential from mathematical physics. It's available here:

http://www.emis.de/journals/SLC/wpapers/s44cartier1.pdf
I eventually figured out how to exactly compute, and implement in Haskell, the "functional" logarithm and exponential of (some) formal power series. So, for example, we can get the formal power series for arcsin as exp(-log(sin)) where log and exp here are the functional versions.

In fact, exp here is just the exponential map from differential geometry, and log is its inverse.

I wrote it up here: http://blog.sigfpe.com/2017/02/logarithms-and-exponentials-of-functions.html
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With the latest tools, it's now possible to change the mouth and facial expression of a person in real time, and to edit audio to completely change what a person is saying. For the facial expression, the new facial expressions are taken from an actor, and for the voice changes, you can simply type the words you want the person to say. Watch out for very realistic fake news.

https://mathbabe.org/2016/12/23/creepy-tech-that-will-turbocharge-fake-news/
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