A Quote from Tesla which is very predictive in one way, and perhaps not in another way

 Nikola Tesla, famous inventor, who lived 1856--1943 said the following:

When wireless is perfectly applied the whole earth will be converted into

a huge brain, which in fact it is, all things being particles of a real

and rhythmic whole. We shall be able to communicate with one another

instantly, irrespective of distance. Not only this but through television

and telephony, we shall see and hear one another as perfectly as though

we were face to face, despite intervening distances of thousands of miles;

and the instruments through which we shall be able to do this will be

amazingly simple compared with our present telephone. A man will be able to

carry one in his vest pocket.

The `vest pocket' at the end really impressed me.

By `a man will be able to carry one...' I don't know if he mean all people or if he actually 

meant that women would not need such a device. If that is what he meant then,

while high marks for tech-prediction, low marks for social-prediction. 

This quote is SO right-on for technology that I offer the following challenge: Find other quotes from year X that were very predictive for year X+Y for a reasonably large Y.

Remembering 2000

FOCS 2000 took place in Redondo Beach, just south of Los Angeles, November 12-14. Certainly some great results such as the Reingold-Vadhan-Wigderson Zig-Zag Graph Product Expander construction that would lead to Omer Reingold's Undirected Connectivity in Log Space. Mostly though I remember the discussions about the presidential election held the week before and whether we might find out our next president during the conference. Spoiler alert: We didn't. 

Consider the following viewpoints for a person X

1. Did X support Bush or Gore?

2. Did X interpret the rules of the election that Bush won or Gore won?

These should be independent events. Your interpretation of the rules should not depend on who you supported. But in fact they were nearly perfectly correlated. Whether you were a politician, a newspaper editorial page writer, a supreme court justice, a computer scientist or pretty much everyone else, if you supported Gore, you believed he won the election and vice-versa. Everyone had their logic why they were right and I'm sure my readers who remember that election still believe their logic was correct. 

As this upcoming election gets messy, as it already has, take care with trying to justify your desired endgame by choosing the logic that makes it work. Would you use the same logic if the candidates were reversed? Everyone says "yes" but it's rarely true. Just like Mitch McConnell, you'll just find some excuse why the opposite situation is different. Trust me, my logic is impeccable. 

Baseball can go on forever, it doesn't just seem that way

 Most games have some way to make sure they cannot go on forever.

1) Chess: I had thought there was a 50-move rule and a 3-times-same-position rule, but its a byte more complicated than that, see here. There is also a chess clock. Suffice to say, Chess can never on forever (though it may seem like it does). 

2) NIM: Eventually all of the stones are gone. There may be more complicated versions where you can add some stones, but in those versions I suspect that there is some parameter that goes to 0.

3) Basketball, Football, Hockey, Soccer: These all have a clock so they are time limited. For overtime there are also rules that make sure the game cannot go on forever. Or maybe its just very rare: what if the Superb Owl (spelled that way to avoid lawsuits, see here) is tied 0-0 at the end of the four quarters and goes into overtime and... nobody scores... ever. Could the game go on forever or would the referees declare it a tie? In the regular season there are ties, but in the in the superb owl? Actually this may be more a problem in the playoffs since you need to determine who goes to the next round.

4) Take your favorite game. I would bet dollars to doughnuts (what an odd phrase---see here for more about the phrase) that there is some mechanism to make sure the game ends. An exception that Darling pointed out to me: If in Gin Rummy both players are terrible then the game can go on forever. This is probably true for other games as well and actually makes the question into two questions (a) will a game terminate no matter what the players do, and (b) (not sure how to formalize) will a game terminate if both players are trying to win and are making reasonable moves.

You may have noticed that in item 3 I left out Baseball. There is no clock in baseball. So one way the game can go on forever is to have a tie and extra innings and nobody scores. I think the umpire has the authority to call it a tie. (Incidentally, the shortened baseball season has a new extra inning rule---each inning starts with a runner on second. See here,) When Lance read an earlier version of this post he pointed me to 5 ways a game can go on forever, not counting the example I have later in this post. Here is where Lance found the question and answer (look on the first page under Speed Department for the question, and the very end of the second page for the answer). I also did my own writuep with more details, see here.  Also of interest (though not if you were actually at the game this happened), the record for number of times a player has a foul with 2 strikes is 16, see here. 

 However, I came across an  example more obscure than any of those. 

Here is what happened (and you can see the video of it here, though it really starts about a minute into it. Keep reading- it looks like its another post, but its part of this post: 

From your Digest

Zev Steinhardt

·

July 9, 2019

Studied at Pace University

Has a play ever happened in baseball that was so out of the ordinary that no written umpiring rule at the time covered it?

Back in 2008, the Yankees drafted a pitcher named Pat Venditte. What made Venditte unusual is that he can throw with both hands. In other words, he’s a switch pitcher. When he was drafted, he was assigned to the Staten Island Yankees, a low A ball team.

In his first game (against the Mets farm team, the Brooklyn Cyclones), Venditte came in to pitch. After getting the first two batters out and giving up a single, he then faced Ralph Henriquez, was a switch hitter. What happened next resembled an Abbott and Costello comedy routine. Venditte would put the glove on one hand (he had a specially made glove that could be worn on either hand) and Henriquez would then step across the plate to bat from the other side. Venditte would then switch his glove hand again and Henriquez went back to the other side.

Eventually, after much discussion, the umpires ruled that Henriquez would have to choose a batting side first, before Venditte had to commit. Henriquez was mad and, after he struck out, he slammed the bat against the ground in frustration.

The umpires were, in essence, winging it, because there was no rule to cover the situation. Eventually, the higher ups in baseball did write a rule to cover the situation — the opposite of the umpires’ decision.

An interesting serendipitous number

 Last seek I blogged about two math problems of interest to me here.

One of them two people posted answers, which was great since I didn't know how to solve them and now I do. Yeah! I blogged about that here.

The other problem got no comments, so I suppose it was of interest to me but not others. I was interested in it because the story behind it is interesting, and the answer is interesting.

it is from the paper 

An interesting and serendipitous number by John Ewing and Ciprian Foias, which is a chapter in the wonderful book 

Finite vs Infinite: Contributions to an eternal dilemma

Here is the story, I paraphrase the article (I'll give pointers  later).

In the mid 1970's a student asked Ciprian about the following math-competition problem:

x(1)>0    x(n+1) =  (1 + (1/x(n)))^n. For which x(1) does x(n) --> infinity?

It turned out this was a misprint. The actual problem was

x(1)>0  x(n+1)=(1+(1/x(n))^{x(n)}. For which x(1) does x(n) --> infinity.

The actual math-comp problem  (with exp x(n)) is fairly easy (I leave it to you.) But this left the misprinted problem (with exp n).  Crispian proved that there is exactly ONE x(1) such that x(n)--> infinity. 

Its approx 1.187... and may be trans.

I find the story and the result interesting, but the proof is to long for a blog post.

I tried to find the article online and could not. A colleague found the following:

A preview of the start of the article here

Wikipedia Page on the that number, called the Foias constant, here

Mathworld page on that number here

Most of the article but skips two pages here

When are both x^2+3y and y^2+3x both squares, and a more general question

 In my last post (see here) I asked two math questions. In this post I discuss one of them. (I will discuss the other one later, probably Monday Sept 14.)

For which positive naturals x,y are x^2+3y and y^2+3x both squares?

I found this in a math contest book and could not solve it, so I posted it to see what my readers would come up with. They came up with two solutions, which you can either read in the comments on that post OR read my write up here.)

The problem raises two more general questions

1) I had grad student Daniel Smolyak write a program that showed that if  1\le x,y \le 1000 then the only solutions were (1,1) and (11,16) and (16,11).  (See write up for why the program did not have to look like anything close to all possibly (x,y).)  

Is there some way to prove that if the only solutions for 1\le x,y\le N (some N) are the three given above, then there are no other solutions?

2) Is the following problem solvable: Given p,q in Z[x,y] determine if the number of a,b such that both p(a,b) and q(a,b) are squares is finite or infinite.  AND if finite then determine how many, or a bound on how many.

Can replace squares with other sets, but lets keep it simple for now. 

Two Math Problems of interest (at least to me)

 I will give two math problems that are of interest to me.

These are not new problems, however you will have more fun if you work on them yourself and leave comments on what you find. So if you want to work on it without hints, don't read the comments.

I will post about the answers (not sure I will post THE answers) on Thursday.

1) Let x(1)>0. Let x(n+1) = (  1 + (1/x(n))  )^n. 

For how many values of x(1) does this sequence go to infinity?

2) Find all (x,y) \in N \times N such that x^2+3y and y^2+3x are both squares. 

A well known theorem that has not been written down- so I wrote it down- CLIQ is #P-complete

(The two proofs that CLIQ is #P-complete that I discuss in this blog are written up by Lance and myself and are here. I think both are well known but I have not been able to find a writeup,so Lance and I did one.)

 I have been looking at #P (see my last blog on it it, here, for a refresher on this topic) since I will be teaching this topic in Spring 2021.  Recall

#SAT(\phi) = the number of satisfying assignments for phi

#CLIQ((G,k))= the number of cliques of size \ge k of G

#SAT is #P-complete by a cursory look at the proof of the Cook-Levin theorem.

A function is #P-complete if everything else in #P is Turing-Poly red to it. To show for some set A 

in NP, #A is #P-complete one usually uses pars reductions. 

I wanted a proof that #CLIQ is #P-complete, so I wanted a parsimonious reduction from SAT to CLIQ (Thats a reduction f: SAT--> CLIQ such that the number of satisfying assignments of phi equals the number of cliques of size \ge k of G.)

I was sure there is such a reduction and that it was well known and would be on the web someplace. So I tried to find it.

ONE:

I tracked some references to a paper by Janos Simon (see here)  where he claims that the reduction of SAT to CLIQ  by Karp (see here) was pars.  I had already considered that and decided that Karps reduction was NOT pars.  I looked at both Simon's paper and Karp's paper to make sure I wasn't missing something (e.g., I misunderstood what Simon Says or what Karp ... darn, I can't think of anything as good as `Simon Says'). It seemed to me that Simon was incorrect. If I am wrong let me know.

TWO

Someone told me that it was in Valiant's  paper (see here). Nope. Valiant's paper shows that counting the number of maximal cliques is #P-complete. Same Deal Here- if I am wrong let me know. One can modify Valiant's argument to get #CLIQ #P-complete, and I do so in the write up. The proof is a string of reductions and starts with PERM is #P-complete. This does prove that# CLIQ is  #P-complete, but is rather complicated. Also, the reduction is not pars.

THREE 
Lance told me an easy pars reduction that is similar to Karp's non-pars reduction, but it really is NOT Karp's reduction. Its in my write up. I think it is well known since I recall hearing it about 10 years ago. From Lance. Hmmm, maybe its just well known to Lance.

But here is my question: I am surprised I didn't find it on the web. If someone can point to a place on the web where it is, please let me know. In any case, its on the web NOW (my write up) so hopefully in the future someone else looking for it can find it.

Sharp P and the issue of `natural problems'

 #P was defined by Valiant as a way to pin down that the PERMANENT of a matrix is hard to compute.

The definition I give is equivalent to the one Valiant gave.

g is in #P if there exists p a poly and B in P such that

g(x) = | { y : |y| = p(|x|) and (x,y) \in B } |

A function f is #P-complete if g is in #P and for all g in #P,  f is poly-Turing reducible to g.

#SAT is the function that, given a formula, returns the number of satisfying assignments. It is #P-complete by looking at the proof the Cook-Levin Theorem. The reduction of f to #SAT only makes one query to #SAT. A common way to show that #A is #P-complete is to show that SAT \le A with a reduction that preserves the number of solutions. 

Valiant proved that PERM was #P-complete (his reduction only used 1 call to PERM).

There are problems in P whose #-version is #-P complete: Matching and DNF-SAT are two of them.

Notice that I defined #SAT directly, not in terms of a poly p and a set B as above. Here is why: if you use poly p and set B one can do obnoxious things like: 

SAT = { phi : exists yz 2n-bits long such that phi(y)=T and z is prime }

The # version of this definition is not really what I want (though I am sure its #P-complete).

Valiant (see here and here) and Simon (see here) showed that  for many known NPC-problems A, #A is #P-complete. They meant NATURAL problems. Is it true for all natural NP-complete problems?

Unfortunately the statement `All NATURAL NPC problems give rise to #P-complete functions' is hard (impossible?) to state rigorously and hence hard (impossible?) to prove. 

1) Is there a natural A in NP such that #A is NOT #P-complete (under assumptions)?

2) Are there any theorems that show a large set of NPC problems have #P counterparts? Or are we doomed to, when we want to show some #A is #P-complete, come up with a new proof?

3) Can one PROVE there are NPC problems A such that #A is NOT #P-complete? (under assumptions).