Slicing the Hypercube

Here's a neat result I heard about at virtual Dagstuhl last week, a new lower bound on the number of hyperplanes that cuts all the edges of a hypercube.

A n-dimensional hypercube has 2ⁿ vertices corresponding to the binary strings of length n. Edges are between two vertices that differ in exactly one bit, for a total of (n/2) 2ⁿ edges. Hypercubes played a large role in Hao Huang's recent proof of the sensitivity conjecture. 

A hyperplane is just a linear inequality in x₁,…,x_n the bits of the string corresponding to a vertex. An edge is cut if the inequality is true for one vertex and false for the other.

With n hyperplanes you can cut all the edges in two very different ways. 

• The hyperplanes x_i > 0 for each i. These are n orthogonal hyperplanes.
• The hyperplanes x₁+…+x_n > i for each i from 0 to n-1. These are n parallel hyperplanes.

Do you need n hyperplanes to cut all the edges? Mike Paterson found 5 explicit hyperplanes that cuts all the edges of a 6-dimensional hypercube (see survey by Mike Saks). Scaling that up means you need 5n/6 hyperplanes to cut an n-dimensional hypercube. That remains the best known upper bound.

For the lower bound, in 1971 Patrick O'Neil showed that any hyperplane can cut at most O(n^0.5) fraction of the edges (sharp by the hyperplane x₁+…+x_n > n/2). Thus you need at least O(n^0.5) hyperplanes which for 50 years was the best known bound.

Gil Yehuda and Amir Yehudayoff have given a new lower bound of O(n^0.57). The paper gives a O(n^0.51) bound but Yehudayoff said in a talk last week the bound has been improved.

Yehudayoff didn't go into much details in his 20 minute talk but the proofs uses geometry, probability and antichains. 

The result has some applications to complexity, namely you need at least n^1.57 wires in a depth-two threshold circuit for parity. But the main fun is the question itself, that we finally made progress and there is still a long way to go.

The key to my Taylor series problem: Buddy can you spare a penny, nickel, dime, or quarter

 In my last blog post I posed a question about finding the coeff of x^100 in a particular Taylor Series. The question and answer are given  here:

The key to the problem was to recognize that it was asking how many ways you can make change of a dollar using pennies, nickels, dimes, and quarters. This can be done by hand (its 242). 

1) Someone who I gave the problem to solved it by available software, but when he saw the answer was 242 he realized how to do it via making change.

2) How hard would this problem be to do completely by hand- say as a Putnam Problem? Its hard for me to say since I started with the trick and then found the problem. 

3) Is this a well known trick? 

A Taylor Series Problem

 I post a problem today and its solution on Thursday.

Comments are fine, though if you don't want to get a hint, don't read them. 

Find the coefficient of x¹⁰⁰ in the Taylor series for the rational function which has 

numerator 1 

and denominator

x⁴¹ - x⁴⁰ - x³⁶ + x³⁵ -x³¹ + x³⁰ + x²⁶ - x²⁵- x¹⁶ + x¹⁵ + x¹¹ - x¹⁰ + x⁶ - x⁵ -x+1

For better readability see my pdf file with the problem in it here

Is there a clever way to do the problem?  If the way to do it was to actually do the Taylor series then 

1) I wouldn't post it

2) I probably could not do it (or it would take too long to bother)  though maybe there are freely available programs that could. 

So yes, there is a clever solution. At least I think it's clever. 

I read the news today oh boy

I'm absolutely elated that Lázló Lovász and Avi Wigderson won the Abel Prize. More from the New York Times, Quanta Magazine and Gil Kalai. Another example of how complexity theory and theoretical computer science has reached the upper echelons of mathematics.

I'm horrified of the spa shootings in Atlanta. I've driven by those spas many times when I lived there. 

I'm saddened by the closing of Mills College in Oakland, California. I lived in a dorm Berkeley rented at Mills College during my crazy year there.

I've got mixed emotions about the death of James Levine. What he did musically with the Metropolitan opera and orchestra was nothing short of incredible. What he did to the young people he abused is inexcusable. I remember watching a Met taped videocast of Die Walküre with my daughter, enjoying the production but telling her "Do you realize we are watching an opera composed by an anti-Semite and conducted by a child molester?"  Can you separate art from artist?

I didn't understand The Art Market before the NFT sale, and I understand it less now

 (This post is a sequel to my post from Feb 13, 2007 which you can find here. While the gap from 2007 until 2021 seems large, its not as long as the gap between Knuth Vol 3 and Knuth Vol 4, nor as long as the gap between Stan Freberg Vinyl Record History of America Part I and his CD History of America Part 2, both novelty records, and quite good.)

My 2017 post was about people posting a clip on youtube and calling it `rare footage of...' 

My point was: How can something be rare if its on you tube?

I also pondered: if someone can make a REALLY REALLY GOOD copy of the Mona Lisa, that is at talent that should be respected and such a person should be able to sell it for about the same price as the original (not sure if I want the copy to be worth A LOT or the original to be worth LESS than it is.)

IF Art is to-look-at-cause-its-pretty then it should not matter who the artist is. 

IF Art is an investment then that could be risky since it does not have intrinsic value. 

IF Art is neither to-look-at or an investment then... What is it? We'll see below that one answer might be Bragging Rights. 

This is NOT a RANT, this is an admitance of my lack of understanding. (Spell check things admitance is not a word. Maybe its admitence. No, Hmmm.) 

And now there is a new wrinkle in all of this: 

69 million for a NFT (Non-Fungable Token) of an art work:story here

1) What is the buyer getting? Bragging rights?

2) Can anyone SEE the art but doesn't OWN it? I don't know..

3) If someone hacked in and got a perfect copy of the art and posted it on a website, would that be theft? Nothing was taken. 

4) In the normal art world does it happen that prices go DOWN because people wake up and say WHAT? Why did I EVER think that White on White was worth 10 million dollars? 

5) Might this happen here also? 

6) Is My Feb 13 blog, which is in a diff format (or I didn't know how to use the blogger interface then) going to one day be worth something?

Cake Cutting in Overtime

There's a new proposal out of Baltimore for a new way to handle overtime in National Football League games. This post is about American Football, soccer has its own overtime issues we won't talk about here.

Instead of randomly choosing who gets the ball, which gives an advantage to the team on offense, the Raven's coach John Harbaugh suggests a "spot and choose" rule based on cake cutting. One team picks a starting position and the other team decides whether to be on offense or defense.

Sounds intriguing but there's a problem. In cake cutting, if you cut off a crumb, everyone would definitely choose the rest of the cake. But in football suppose the spotting team chose the offensive's team one-yard line (99 yards needed for a touchdown). For spot and choose to work the one-yard line would have to be an obvious choice for defense. But many teams might still choose starting at the one-year line on offense. There's a chance you can march down the field and if not you can always punt it away. So the team that gets to choose whether to be on offense could get an advantage no matter what the spotting team did.

I still like the idea of "spot and choose". Maybe you let the first team choose not only the yard line but what down to start. Because no one would start at their one yard line at 4th down.

Are there variations for spot and choose in other sports? I like using game theory to figure out how to play actual games. 

When do I need to warn about Spoilers?

In a recent post here I mentioned in passing a plot point from the last season of The Big Bang Theory. Note that the last season was in 2019.  WARNING- do not read that post if you are watching The Big Bang Theory and do not want a plot point revealed. 

Someone who perhaps thinks Lance and I are the same person (are we? See here) left Lance a tweet complaining about the spoiler. At least I think they are complaining. The tweet is in Spanish and its here.

Either

1) Some country is two years behind America on showing The Big Bang Theory. 

2) The person who tweeted has them on DVR (or something like that) and is watching them a few years after they air (I watched Firefly on a DVD I borrowed from a friend 10 years after it went off he air. Ask your grandparents what a DVD used to be.) 

3) They are kidding us and making fun of the notion of spoilers.

This raises the question: When is it okay to post spoilers without warning? A few random thoughts:

1) ``Don't tell me who won the superb owl! I have it on tape and want to watch it without knowing who won!''  This always seemed odd to me.  Routing for events to happen that have already happened seems weird to me. When I was 10 years old I was in New York listening to a Knicks-Celtics Basketball game on the radio and during halftime I accidentally found a Boston radio station that had the game 30 minutes later (I did not realize that the channel I was on originally was 30 minutes behind). So I heard how the game ended, then switched back listening to a game knowing how it would end. I didn't route for my team (the Knicks, who lost) but it just felt very weird listening to it. If I had thought of it I might have noticed how the different broadcasts differ and got a paper out of the data, but as a 10 year old I was not thinking about how to pad my resume quite yet. 

2) I like seeing a mystery twice- first time I don't know who did it, second time I do but can look for clues I missed the first time.

3) I would have thought 2 years after a show is off the air its fine to spoil. But... maybe not.

4) It also matters how important the plot point is. I didn't think the plot point I revealed was that important. 

5) Many TV shows are predictable so I am not sure what `spoiler' even means. If I said to Darling:

 The bad guy is an unimportant character we meet in the first 10 minutes.

that does not show I've seen it before. It shows that I am a master of TV-logic.

6) With Arc TV shows this is more of a problem. While it was possible to spoil an episode (Captain Kir will survive but Ensign Red Shirt will bite the dust) it was impossible to spoil a long-term arc. TV has gotten to complicated. And I say that without having watched Game of Thrones.