What is a natural question? Who should decide?

(Thanks to Timothy Chow for inspiring this post.)

My survey on Hilbert's Tenth Problem(see  here) is about variants of the problem. One of them is as follows: 

For which degrees d and number-of-vars n, is Hilbert's tenth problem decidable? undecidable? unknown? 

 I wondered why there was not a website with this information. More generally, the problem didn't seem to be getting much attention. (My survey does report on the attention it has gotten.) 

I got several emails telling me it was the wrong question. I didn't quite know what they meant until Timothy Chow emailed me the following eloquent explanation:

-----------------------------------

One reason there isn't already a website of the type you envision is that from a number-theoretic (or decidability) point of view, parameterization by  degree and number of variables is not as natural as it might seem at first glance. The most fruitful lines of research have been geometric, and so geometric concepts such as smoothness, dimension, and genus are more natural than, say, degree. A nice survey by a number theorist is the book Rational Points on Varieties by Bjorn Poonen. Much of it is highly technical; however, reading the preface is very enlightening. Roughly speaking, the current state of the art is that there is really only one known way to prove that a system of Diophantine equations has no rational solution.

----------------------------------

AGAINST THE NUMBER THEORISTS VIEWPOINT:

1) ALICE: Why are you looking for your keys under the lamppost instead of where you dropped them?

   BOB: The light is better here.

2) I can imagine the following conversation:

BILL: I want to know about what happens with degree 3, and number of variables 3.

MATHPERSON: That's the wrong question you moron. The real question is what happens for fixed length of cohomology subchains.

BILL: Why is that more natural?

MATHPERSON: Because that is what we can solve. And besides, I've had 10 papers on it.

FOR THE NUMBER THEORISTS VIEWPOINT

1) They are working on really hard problems so it is natural to gravitate towards those that can be solved.

2) I suspect that the math that comes out of studying classes of equations based on smoothness, dimension, genus is more interesting than what comes out of degree and number of vars. Or at least it has been so far. 

META QUESTION

Who gets to decide what problems are natural?

People outside the field (me in this case) are asking the kind of questions that a layperson would ask and there is some merit to that.

People inside the field KNOW STUFF and hence their opinion of what's interesting to study has some merit. But they can also mistake `I cannot solve X' for `X is not interesting'

Does the university matter?

As we come out of a pandemic with online teaching and research collaborations, how much do we actually need the university?

Theoretical research in computer science, math and elsewhere it hardly slowed down with everyone hunkered down at home, and when it did it was more because of supervising kids with their on-line learning than being away from the university. Collaborating with those around the world was basically the same as collaborating with your colleagues on campus.

Many courses, especially the larger ones, worked about as well on-line as they do in person.

The pandemic is accelerating changes already in place. Before say 1960, the fastest travel for the masses was on train and boats and fast communication limited to expensive phone calls. You needed strong colleagues, a strong library and a strong support staff to be a successful academic. Traveling to meet colleagues and attend conferences was a luxury that few could do often.

The 60's gave us air travel though it wasn't until the 90's that we could readily send academic papers electronically. It's really only recently that we have the infrastructure to allow high-quality teaching and research collaboration online.

Suppose universities now just disappeared. Professors would be free agents, supported by grants and tuition from students taking their on-line courses and consulting. Students would pick and choose courses from the best instructors, their "transcript" being recorded on a blockchain-like database. PhD students would be more of an apprenticeship, a low salary in exchange for personalized mentoring from a professor. 

For the experimental sciences, there would be a set of national labs that professors could join.

Startups would create apps to enable all these things, professors would just become yet another piece of the gig economy. Superstar academics can pull in large salaries, the rest would struggle to make a decent wage--not that different from actors and musicians. 

Don't worry, universities aren't going anywhere soon. Or are they?

Emerging from the Pandemic

The City of Chicago yesterday agreed with the latest CDC guidelines that those of us fully vaccinated no longer have to wear masks in most settings. Lollapalooza, the big Chicago music festival, will be held in full capacity this summer. There are still some restrictions but barring any surprise setbacks should be mostly gone by the Fourth of July.

It's not appropriate to call the pandemic over. Too many countries are suffering and lack good access to vaccines or strong health care. But in most of the US vaccinations are readily available and it really feels like we are putting the pandemic in the past.

Variants that beat the vaccines could emerge. Vaccines could become ineffective over time. Too many still need to be vaccinated and too many don't want to do so. Typically problems start small and quickly get big by exponential growth but likely with enough warning if we need boosters or extra precautions. And all those who cut in line to get shots early are now my canaries for the effects wearing off.

What will this post-pandemic world look like? Continued virtual meetings from people who don't want to spend the 10-15 minutes to get across campus or to come to campus at all. A bigger move to casual dress, even in the corporate world. No more snow days. Changes in ways we won't expect. 

We all have our different attitudes but I'm ready to move on. Time to start the "after times".

Why do countries and companies invest their own money (or is it?) in Quantum Computing (Non-Rhetorical)

 There have been some recent blog posts by Scott (see here) and Lance (see here)  about the hype for SHORT TERM APPLICATIONS of Quantum Computing, which they both object to. 

I have a question that has been touched on but I want to get it more out there.

PREAMBLE TO QUESTION:  The following scenarios, while distasteful, do make sense: 

a) A researcher on their grants exaggerates or even out-right lies about the applications of their work. 

b) A journalist in their articles exaggerates or even out-right lies about the applications of the science they are covering.

c) A company exaggerates or even out-right lies about the applications of their project to a venture capitalist or other kind of investor. 

QUESTION: 

Why does a company or country invest THEIR OWN MONEY into Quantum Computing which is unlikely to have a short term profit or benefit? Presumably they hire honest scientists to tell them the limits of the applications in the short term. 

ANSWERS I HAVE GOTTEN FROM ASKING THIS

1) QC might be able to do something cool and profitable, like factoring, or simulating physics quantum experiments or something else, in the short term. Quantum Crypto is already happening, and that's a close cousin of Quantum Computing. 

2) QC might be able to do something cool and profitable (like in (1)) in the long term, and both companies and countries think they will be around for a long time. (For a list of America's 10 oldest companies see here.) 

3) The company or country is in this for the long term, not for a practical project, but because they realize that doing GOOD SCIENCE is of a general benefit (this might make more sense for a country than a company). And funding Quantum Computing is great for science. 

4) Bait and Switch: The company claims they are doing Quantum to attract very smart people to work with them, and then have those smart people do something else.

5) (this is a variant of 1) While Quantum Computing may not have any short term applications, there will be classical applications INSPIRED by it (this has already happened, see here).

6) Some of these companies make money by applying for GRANTS to do QC, so its NOT their money. In fact, they are using QC to  GET money.

7) For a country its not the leaders money, its that Taxpayer's money- though that still leaves the question of why spend Taxpayer money on this and not on something else.

8) For a company its not their money- its Venture Capitalists  and others (though for a big company like Google I would think it IS their money). 

9) The scientists advising the company or country are giving them bad (or at least self-serving) advice so that those scientists can profit- and do good science. So this is a variant of (3) but without the company or country knowing it. 

10) In some countries and companies group-think sets in, so if the leader (who perhaps is not a scientist) thinks intuitively that QC is good, the scientists who work for them, who know better, choose to not speak up, or else they would lose their jobs...or worse. 

11) For countries this could be like going to the moon: Country A wants to beat Country B to the moon for bragging rights. Scientists get to do good research even if they don't care about bragging rights. 

12) (similar to 11 but for a company) If a company does great work on QC then it is good publicity for that company. 

13) Some variant of the greater fool theory. If so, will there be a bubble? A bail-out? 

Cryptocurrency, Blockchains and NFTs

 I first wrote about bitcoin in this blog ten years ago after I gave a lecture in a cryptography class I taught at Northwestern. Two years later I had a follow-up post, noting the price moved from $3 to $1000 with a market cap of about $11 Billion. My brother who thought they were a scam back then has since become a cryptocurrency convert. The bitcoin market cap is now over a trillion dollars and other cryptocurrencies are not far behind. No longer can we view cryptocurrencies as simply a neat exercise in applied cryptography now that it has serious value.

The main uses of cryptocurrencies are for speculation or illegal activities, such as drug sales, ransoms, money laundering and tax evasion. Sure you can buy a Tesla with bitcoins but that's more of a gimmick. Cryptocurrency spending is simply too slow, expensive and volatile right now to replace other methods of electronic payment. 

Non-fungible tokens (NFTs) truly puzzle me. They are just a digital certificate of authentication. What could you do with them you couldn't do with docusign? Collectibles of publicly available digital goods is a fad already fading.

I'm not a fan of a fiat currency governed by strict rules not under governmental control. Bad things could happen. However thinking of cryptocurrencies and the blockchain technology that underlies them have brought up real needs for our digital world.

• An easy way to pay online without significant fees, expenses or energy consumption.
• An easy and cheap way to transfer money between different countries.
• A distributed database to allow tracking of supply chains, credentials and financial transactions for example. I see less a need to make these databases decentralized.
• A need, for some, to have a digital replacement for the anonymity of cash.
• People need something to believe in once they have given up believing in religion and a functioning democracy. 

Don't change your investing habits based on anything I write in this post. Speculation and illegal activities are powerful forces. Or it could all collapse. Make your bets, or don't.

Note: Since I wrote this post yesterday, Elon Musk tweeted that Tesla will no longer accept bitcoins, and the bitcoin market cap has dropped below a trillion.

Trump, Facebook, and ComplexityBlog

 I care about the Facebook decision to ban Trump, but I do not have a strong opinion about it. I have heard arguments on both sides now, from up and down, and still somehow... I don't know how I feel. So instead of posting my opinion I post other opinions and my opinion of them.

1) Facebook is a private company. If they want to have liberal bias or a free for all or whatever then  it is not the governments place to interfere. If enough people don't like what they see then they will lose customers. The invisible hand of the market will regulate it enough. Libertarians and honest small-gov republicans might believe this. On a personal level, I don't want someone else telling Lance and I that we can't block some comment; however, for now, more people use Facebook then read Complexity Blog. 

2) Facebook is a private company but they need to follow standard business practices of having their uses sign an agreement and stick to it. Since the user signed the agreement, Facebook need only stick to that agreement. This is problematic in that (1) if the agreement is not that rigorous then Facebook can be arbitrary and capricious, but (2) if the agreement is to rigorous then people can game the system. Imagine if Lance and me had  rule that you could not use profanity in the comments. Then someone could comment 

People who think P vs NP is ind of ZFC can go Fortnow themselves. They are so full of Gasarch.

 (Something like this was the subplot of an episode of The Good Fight)

3) Facebook is so big that it has an obligation to let many voices be heard, within reason. This could lead to contradictions and confusions:

a) Facebook cannot ban political actors. What is a political actor? (Jon Voight is pro-trump and Dwayne ``The Rock'' Johnson is anti-trump, but that's not what I mean.) High level people in the two main parties qualify (how high level?). IMHO third parties (Libertarian and Green come to mind) need the most protection since they don't have as many other ways to get out their message and they are serious. (I wonder if Libertarians would object to the Government  forcing Facebook to not ban them). What about the Surprise Party or the Birthday Party (which did have a platform see here). And what about people running for Mayors of small towns (much easier to do now with Facebook)? Should just running be enough to ban banning? 

b) Facebook can ban posts that are a threat to public health and safety. I am thinking of anti-vaxers and insurrectionists, though I am always wary of making them free speech martyrs. 

c) Fortunately a and b above have never conflicted. But they could. I can imagine a president who has lost an election urging his followers to storm the capitol. Then what should Facebook do?  (ADDED LATER- A commenter points to a case where a and b conflicted that is not the obvious case.) 

4) Facebook is so big that it has an obligation to block posts that put people in danger. This may have some of the same problems as point 3---who decides? 

5)  Facebook is so big and controls so much of the discourse that it should be heavily regulated (perhaps like a utility).  This has some of the same problems as above- who decides how to regulate it and how?

6) As a country we want to encourage free speech and a diversity of viewpoints. There are times when blocking someone from posting may be better for free speech then letting them talk. When? When that person is advocating nonsensical views that stifle the public discussion. But I am talking about what the country should want. What do they want? What does Facebook want? Does either entity even know what they want? These are all ill defined questions. 

7) Facebook is a monopoly so use Anti-Trust laws on it. Anti-Trust was originally intended to protect the consumer from price-gouging. Since Facebook is free this would require a new interpretation of antitrust. Judicial activism? The Justices solving a problem that the elected branches of government are currently unable to solve? Is that a bad precedent? What does it mean to break up Facebook anyway--- its a network and hence breaking it up probably doesn't make sense (maybe use MaxCut). 

(ADDED LATER- a commenter pointed out that anti-trust is NOT just for consumer protection, but also about market manipulation to kill small innovators.) 

8) Lets say that Facebook and Society and the Government and... whoever, finally agree on some sort of standards. Then we're done! Not so fast. Facebook is so vast that it would be hard to monitor everything. 

9) As a side note- because Facebook and Twitter have banned or tagged some kinds of speech or even some people, there have been some alternative platforms set up. They always claim that they are PRO FREE SPEECH. Do liberals post on those sites? Do those sights  ban anyone? Do they have SOME rules of discourse? I ask non rhetorically. 

Negotiations

So you got an offer to be an assistant professor in the computer science department at Prestigious U. Congratulations! 

Time to negotiate your offer with the chair. Don't be nervous. This shouldn't be adversarial. Both of you have the same goal in mind--for you to come to Prestigious and be successful. 

Let's discuss the different aspects of each package.

Research 

Funds for supporting your research such as equipment, graduate student support, travel and postdocs. Here you should explain what you need to be successful. This will vary by subdiscipline, a systems researcher will need more equipment and students than a theorist. Keep in mind the university is giving you funds for 2-4 years to start your research, after which you are expected to fund your own research via grants.

I don't recommend taking on a postdoc right at the start of your first academic appointment. Postdocs require good mentoring while you need to spend the first year getting your research up and running. If you do ask for postdoc money, ask to have a flexible start time.

Many departments give course reductions to get your research going. I'd suggest asking to spend your first semester teaching a graduate topics course based on your thesis research to pick up some PhD students followed by a semester with no classes to get you research program going.

Salary

This includes actual salary, which is also the base for future raises, and summer salary in the first couple of years. Feel free to ask for more salary, but often these numbers are fixed for new assistant professors. There is more give if you take an academic job later in your career. You could also say something like, "Well if you can't give me more salary maybe you could give me another semester of grad student support?"

Partner

It seems 80% of the time, a job candidate has a partner that needs accommodating. Don't wait until the end of negotiations, bring it up early. The more time we have, the better we can help. Doesn't matter what job they want--we know people and we know people who know people.

Thesis

Many schools won't hire you as an assistant professor if you haven't finished your thesis. Has to do with college rankings work. Don't worry--they will generally give you some other role with the same package until you finish. This might delay your tenure clock though.

Delayed start time

A January start is usually fine with good reason but if you weren't planning to start until the fall of 2022 why are you on the market this year? If you do get the department to hold a position for you, remember you are also making a commitment--this is not an opportunity to try again for something better.

Overall

You may not get all that you want after a negotiation--don't take it personally. You shouldn't necessarily choose the place that gives you the biggest package. It's far more important in the long run that you pick a place where you can best succeed both professionally and personally, and the package is just a small piece of that puzzle.

The Mythical Man-Month, Hen-Day, and Cat-Minute (Fred Brooks Turned 90)

 (Added later: Fred Brooks passed away on November 7, 2022, at the age of 90.)

The Mythical Man-Month is a great book which talks about the (obvious in retrospect) fact that putting more people on a project may slow it down. It was by Fred Brooks who turned 90 in April (he is still alive). It's a good read. I actually read it many years ago when I exchanged books with a Software Engineer I was dating- She lent me The Mythical Man Month which I found interesting, and I lent her What is the name of this book by Smullyan which she found amusing. Did this exchange of books help our relationship? We have now been married for many years, though its not clear if we can trace this to the exchange of books OR to the fact that she had KNUTH Volumes 1 and 3, and I had KNUTH Volume 2. 

 Fred Brooks: You have my thanks and of course Happy Birthday!

When I read The Mythical Man-Month  I was reminded of a math problem I heard as a kid: 

If a hen-and-half lays an egg-and-a-half in a day-and-a-half then how many eggs can seven hen lay in seven days? 

My answer: if (3/2) hens lay (3/2) eggs in (3/2) days then that's 2/3 of an egg per hen-day, so the answer is 

49* 2/3 = 32 and 2/3 eggs.

It did not bother me one whit that (1) you can't have 2/3 of an egg, and (2) Just like adding more people might slow down a project, adding more hens might end up being a bad idea-- especially if they are all crowded into the same chicken-coop and hence don't feel much like laying eggs.

Who was the first person to note that adding more people or hens might be a bad idea? I do not know, but here is an amusing, yet realistic, article by Mark Twain on what I would call The mythical cat-minute. My advisor Harry Lewis send it to me in the midst of an email exchange about The Mythical Man-Month. He got it from a student of his, Larry Denenberg. Here it is: 

CATS AND RATS

The following piece first appeared in ``The Monthly Packet'' of February
1880 and is reprinted in _The_Magic_of_Lewis_Carroll_, edited by John
Fisher, Bramhall House, 1973.


   If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill
   100 rats in 50 minutes?

   This is a good example of a phenomenon that often occurs in working
   problems in double proportion; the answer looks all right at first, but,
   when we come to test it, we find that, owing to peculiar circumstances in
   the case, the solution is either impossible or else indefinite, and needing
   further data.  The 'peculiar circumstance' here is that fractional cats or
   rats are excluded from consideration, and in consequence of this the
   solution is, as we shall see, indefinite.

   The solution, by the ordinary rules of Double Proportion, is 12 cats.
   [Steps of Carroll's solution, in the notation of his time, omitted.]

   But when we come to trace the history of this sanguinary scene through all
   its horrid details, we find that at the end of 48 minutes 96 rats are dead,
   and that there remain 4 live rats and 2 minutes to kill them in: the
   question is, can this be done?

   Now there are at least *four* different ways in which the original feat,
   of 6 cats killing 6 rats in 6 minutes, may be achieved.  For the sake of
   clearness let us tabulate them:
      A.  All 6 cats are needed to kill a rat; and this they do in one minute,
          the other rats standing meekly by, waiting for their turn.
      B.  3 cats are needed to kill a rat, and they do it in 2 minutes.
      C.  2 cats are needed, and do it in 3 minutes.
      D.  Each cat kills a rat all by itself, and takes 6 minutes to do it.

   In cases A and B it is clear that the 12 cats (who are assumed to come
   quite fresh from their 48 minutes of slaughter) can finish the affair in
   the required time; but, in case C, it can only be done by supposing that 2
   cats could kill two-thirds of a rat in 2 minutes; and in case D, by
   supposing that a cat could kill one-third of a rat in two minutes.  Neither
   supposition is warranted by the data; nor could the fractional rats (even
   if endowed with equal vitality) be fairly assigned to the different cats.
   For my part, if I were a cat in case D, and did not find my claws in good
   working order, I should certainly prefer to have my one-third-rat cut off
   from the tail end.

   In cases C and D, then, it is clear that we must provide extra cat-power.
   In case C *less* than 2 extra cats would be of no use.  If 2 were supplied,
   and if they began killing their 4 rats at the beginning of the time, they
   would finish them in 12 minutes, and have 36 minutes to spare, during which
   they might weep, like Alexander, because there were not 12 more rats to
   kill.  In case D, one extra cat would suffice; it would kill its 4 rats in
   24 minutes, and have 26 minutes to spare, during which it could have killed
   another 4.  But in neither case could any use be made of the last 2
   minutes, except to half-kill rats---a barbarity we need not take into
   consideration.

   To sum up our results.  If the 6 cats kill the 6 rats by method A or B,
   the answer is 12; if by method C, 14; if by method D, 13.

   This, then, is an instance of a solution made `indefinite' by the
   circumstances of the case.  If an instance of the `impossible' be desired,
   take the following: `If a cat can kill a rat in a minute, how many would be
   needed to kill it in the thousandth part of a second?'  The *mathematical*
   answer, of course, is `60,000,' and no doubt less than this would *not*
   suffice; but would 60,000 suffice?  I doubt it very much.  I fancy that at
   least 50,000 of the cats would never even see the rat, or have any idea of
   what was going on.

   Or take this: `If a cat can kill a rat in a minute, how long would it be
   killing 60,000 rats?'  Ah, how long, indeed!  My private opinion is that
   the rats would kill the cat.

Ferrer's Diagrams can be used to prove X theorems about partitions. What is X?

1978: I took an excellent  ugrad course in combinatorics from James C Frauenthal (he sometimes wrote his name as the biniomial cofficient (J choose F))  and he covered Ferrer's diagrams. They are a nice way to prove equalities about types of partitions.   See here for a definition and a few examples. I have this (possibly false) memory that there were LOTS of partition theorems proven nicely with Ferrer's diagrams.

Fast forward to 2021:

2021: My TA Emily  needs a topic to cover in Honors Discrete Math. I have this memory that there were LOTS of theorems about partitions proven with Ferrer's diagrams. We look at many websites on Ferrer diagrams  and  find only TWO examples:

The numb of partitions of n into k parts is the numb of partitions of n into parts the largest of which is k.

The numb of partitions of n into \le k parts is the numb of partitions of n into parts the largest of which is \le k

We DO find many theorems about partitions such as this corollary to the Rogers-Ramanujan theorem:

The numb of partitions of n such that adjacent parts differ by at least 2 is the numb of partitions of n such that each partition is either \equiv 1 mod 5 or \equiv 4 mod 5.

This is a HARD theorem and there is no Ferrer-diagram or other elementary proof. 

SO, I have one MEMORY but the reality seems different. Possibilities:

1) My memory is wrong. There really are only 2 examples (or some very small number).

2) There are other examples but I can't find them on the web. I HOPE this is true--- if someone knows of other ways to use Ferrer diagrams to get partition results, please comment. 

The Million Dollar Sermon

Illinois Tech has one of the greatest origin stories for a university. In 1890 Frank Gunsaulus, a pastor on the south side of Chicago, gave a sermon where he said "If I had a million dollars I would build a school to provide students from all backgrounds meaningful roles in a changing industrial society". Philip Armour, a wealthy industrialist in the congregation, went up to Gunsaulus after the sermon and said, "if you give me five years for this school, I'll give you the million dollars". Thus started the Armour Institute of Technology which after some mergers became the Illinois Institute of Technology.

The "million dollar sermon" really happened, though the exact wording and even the exact year are lost to posterity or, as speculated, hidden in a cornerstone of one of the original campus buildings. 

In 1890 we were in the beginnings of the second industrial revolution, a revolution of communication, transportation, electrification and soon the assembly line. The first revolution happened a century earlier with mechanization and the steam engine. The third revolution was computers and automation, and we are now in the early parts of the fourth industrial revolution, one based on data and information. 

There are many parallels between 1890 and today. Like 1890, the private economy is dominated by a small number of large companies that have an outsized influence in our society. Like 1890, technology is changing faster than we can manage it. Like 1890, many workers are finding their skills quickly outdated.

Today Gunsaulus's words ring truer than ever. We more than ever need to provide students of all backgrounds meaningful roles in a changing technological society. 

An investment puzzle and speculation as to why some think its hard

 This is a Guest Post by David Marcus. He gives a puzzle and its solution, which is interesting, and then speculates as to why some people get it wrong. 

---------------------------------------------------------------------------

THE PROBLEM:

Investing Puzzle or Arithmetic Can Be Useful

The following is an example of investment results that I saw in an

investment newsletter. There are two portfolios that use different

strategies. Both portfolios start with $1 million twenty years ago and

withdraw 5% each year. The idea is that you are retired and withdrawing

money to spend. Not all years are shown in the tables.

Portfolio A

Year   Return  Withdrawal    Balance

2000   15.31%      57,655  1,095,445

2005    1.81%      59,962  1,139,273

2008  -12.65%      51,000    969,004

2009   34.26%      65,049  1,235,936

2010   11.94%      69,175  1,314,331

2015   -2.48%      64,935  1,233,764

2020   10.27%      66,935  1,271,765

Total Withdrawal: 1,685,190

Change in Balance: 27.18%

======

Portfolio B

Year   Return  Withdrawal    Balance

2000   -0.95%      49,524    940,956

2005    3.80%      44,534    846,154

2008  -20.11%      35,922    682,523

2009   18.27%      40,360    766,833

2010   11.57%      42,777    812,764

2015    0.99%      50,767    964,567

2020   13.35%      65,602  1,246,433

Total Withdrawal: 1,425,573

Change in Balance: 24.64%

Portfolio A has a final balance that is 25,000 more than Portfolio B's and

had about 260,000 more in withdrawals. Does the example lend credence to

the Portfolio A strategy being better than the Portfolio B strategy?

---------------------------------------------------------------------------

THE ANSWER:

Investing Puzzle or Arithmetic Can Be Useful: Analysis

Summary: The two portfolios have about the same performance over the 20

years. The difference is mainly due to Portfolio A having a good year or

years near the beginning before much money was withdrawn. The example

merely shows that it is better to withdraw money after a gain rather than

before.

Detailed Analysis:

The scenario is: Start with X = $1 million. Withdraw 5% a year.

Define "gain factor" to be 1 plus the percentage return. For example, if a

portfolio returns 5%, then the gain factor is 1.05.

Let A_j, j = 1, ..., 20, be the gain factors each year for portfolio A.

Let B_j, j = 1, ..., 20 be the gain factors each year for portfolio B.

The final amount in portfolio A is

   F = X * A_1 * 0.95 * A_2 * 0.95 * ... * A_20 * 0.95 .

The final amount in portfolio B is

   G = X * B_1 * 0.95 * B_2 * 0.95 * ... * B_20 * 0.95 .

From the "Change in Balance" values or the balances for year 2020, we see

that F and G are almost the same:

   F = 1.271865 * X,

   G = 1.246433 * X.

But, as we learned in elementary school, multiplication is commutative, so

   F = X * 0.95^20 * \prod_{j=1}^20 A_j,

   G = X * 0.95^20 * \prod_{j=1}^20 B_j.

Since F and G are almost the same, the total gains (product of the gain

factors) for the two portfolios are almost the same, i.e.,

   \prod_{j=1}^20 A_j \approx \prod_{j=1}^20 B_j.

Then what accounts for the big difference in the amounts withdrawn?

Portfolio A must have had some good years near the beginning. (We see in

the tables that Portfolio A did better in 2000 than Portfolio B.) So, all

the example shows is that it is better to withdraw your money after your

gains rather than before.

To take an extreme example, suppose an investment is going to go up 100%

this year. It is better to take your money out at the end of the year

(after the gain) than at the beginning of the year (before the gain). This

is a triviality.

The example tells us nothing useful about the two strategies.

Note: The total gains aren't exactly the same, but the timing of the yearly

gains is what is driving the results. We have (rounding off)

   ( F - G ) / 0.95^20 = 70942.81 .

So, if there had been no withdrawals, the difference in the portfolio

balances would have been about $71,000, much less than the $260,000 +

$25,000 difference with the withdrawals.

---------------------------------------------------------

WHY IS THIS HARD FOR PEOPLE?

Many people have trouble with this puzzle. The difficulty may be that such

people don't make a mental model (or a written model) of the process that

is producing the balance. If you write down (or have in your head) a

formula for the balance, then you see that the gain factors are independent

of the withdrawal factors. That is, we could withdraw more or less money,

or even deposit money, without affecting the gain factors we would use in

the model. This then leads us to consider the gain factors on their own,

and to recognize that the gain factors are the true measures of how the

strategies perform.

Ordering Beauty

First, congratulations to fellow complexity theorist and blogger Scott Aaronson for receiving the 2020 ACM Prize in Computing for "groundbreaking contributions to quantum computing". The prize is ACM's highest honor for mid-career researchers. Well deserved! 

Now back to our regularly scheduled post...

Every freshman at Cornell back in 1981 had to take two seminar courses, basically one-shot courses in an usually humanities area which required no prerequisites but lots of writing. I took my first course in philosophy. The instructor, a PhD student, at one point described his research, a philosophical argument that there is an intrinsic total ordering of beauty, say that Beethoven would always sit above the Beatles, no matter the beholder. I didn't believe him then and still don't today. A few months ago the Washington Post ran a story with the same theme entitled Maradona was great, and maybe the greatest. Can we make similar claims about artists?

Somehow we have this belief when it comes to conference submissions, that there is some perfect ordering of the submissions and a good PC can suss it out. That's not really how it works. Let's say a conference has an accept rate of 30%. Typically 10% of the submissions are strong and will be accepted by any committee. About half the submissions are just okay or worse and would be rejected. The other 40% of the submissions will be chosen seemingly randomly based on the tastes of the specific members of the program committee. Experiments in the NeurIPS and ESA conferences have bourn this out. 

Why not make the randomness explicit instead of implicit? Have the PC divide the papers into three piles, definite accepts, definite rejects and the middle. Take the third group and randomly choose which ones to accept. It will create a more interesting program. Also randomness removes biases, randomness doesn't care about gender, race and nationality or whether the authors are at senior professors at MIT or first year grad students at Southern North Dakota. 

We put far too much weight on getting accepted into a conference given the implicit randomness of a PC. If we make the randomness explicit that would devalue that weight. We would have to judge researchers on the quality of their research instead of their luck in conferences.

Given that conferences, especially the virtual ones, have no real limits on the number of papers and talks, you might say why not just accept all the papers in the middle. Works for me.

Is the following reaction to getting the first COVID shot logical?

 Alice works at a charity that puts together bag and box lunches for children.

They all wear masks and they are 12 feet apart and very careful, and nobody there has gotten COVID.

Then Alice gets here first COVID shot and says:

I am not going to work for that charity until I have had my second shot and waited  4 weeks so I am immune. 

She is really scared of getting COVID NOW that  she is on the verge of being immune. 

Is that logical? She was not scared before. So does it make sense to be scared now? I see where she is coming from emotionally, but is there a logical argument for her viewpoint? I ask nonrhetorically.

bill g. 

Quantum Stories

Scott Aaronson wrote last month about the hype over quantum computing. I'd thought I'd drop a few stories.

I was once asked to review a grant proposal (outside the US) that claimed it would find a quantum algorithm for NP-hard problems. I wrote a scathing review but the grant was funded because I failed to prove that it was impossible. I replied that they should fund my research to teleport people from Chicago to Paris because they couldn't prove I couldn't do it. I never got a response.

I was at an NSF sponsored meeting on quantum computing. I suggested, as a complexity theorist, that we need to explore the limits of quantum computing. A senior researcher said we shouldn't mention that in the report or it might hurt our chances of funding the field if they think quantum computing might not be a complete success.

I went to a Microsoft Faculty Research Summit which had a big focus on quantum computing. I complained of the quantum computing hype. My friends in the field denied the hype. Later at the summit a research head said that Microsoft will solve world hunger with quantum computing.

I was meeting with a congressional staffer who had worked on the National Quantum Initiative which coincidentally was being announced that day. I said something about high risk, high reward. He looked shocked--nobody had told him before that quantum computing is a speculative technology.

Quantum computing has generated a large number of beautiful and challenging scientific questions. Thinking about quantum has helped generate classical complexity and algorithmic results. But quantum computing having a real-world impact in the near or mid-term is unlikely. Most scientists I know working directly in quantum research are honest about the limitations and challenges in quantum computing. But somehow that message is not often getting to the next layers up, the policy makers, the research managers, the university administrators, the media and the venture capitalists. 

But who knows, maybe some quantum heuristic that doesn't need much entanglement will change the world tomorrow. I can't prove it's impossible.

Do any NP-reductions use deep mathematics? Non-rhetically

BILL: Lets say we didn't know that FACTORING NPC --> NP=coNP.

then what direction would we think Factoring in P or NPC? 

STUDENT: In P. After all, Number Theory is a deep subject and I can imagine some deep Theorem in it leading to FACTORING in P. 

BILL: That cuts both ways. I can imagine some deep theorem in NT being the key to showing 

FACTORING poly-reduces to  SAT

STUDENT: 

Deep mathematics HAS been used for algorithms. Factoring algs is one example, The graph minor theorem yielding MANY problems in P is another. Can you give me an example where deep mathematics has been used for an NPC reduction?

BILL: Oh. Gee. I do not know of any. 

STUDENT: If only you had some mechanism to ask the theory community. Maybe you could call it a web log, or weblog.

BILL: If only...

QUESTIONS

1) Are there any NPC reductions that use deep math? (I realize that the phrase `deep math' is not well defined,)

2) Are there other reductions that use deep math?

3) If P NE NP then: 

For all epsilon there is no approx-alg for MAX3SAT which yields  \ge  (7/8 + epsilon)OPT

For all delta  there is no approx-alg for CLIQ which yields > n^{-delta} OPT

No approx-alg for SET COVER which yields \ge (ln n + o(1))OPT. 

All of these proofs use  the PCP machinery or something similar. My impression is that the original PCP theorem, while long, hard, and impressive, did not use deep math. I have a vague memory of some paper or PhD thesis stating that the ONLY theorem needed was that a poly of degree d over a finite field has \le d roots. 

However to get the optimal lower bounds seemed to use some deep math. But I am more ASKING than telling. 

Want to Buy a Theorem?

This is embarrassing to admit but after a few badly timed trades on GameStop options I find myself a bit tight on money. To raise some cash, I reluctantly decided to sell one of my prized possessions, one of my theorems.

For Sale: Boolean formula satisfiability cannot be solved in both logarithmic space and quasilinear time. For a more formal and slightly more general statement and a proof, see this paper.

Bidding starts at 12 BTC (about $705,000). 

The winning bid, upon verified payment, will receive:

• The ability to give the theorem the name of your choice such as your own name, your best friend's mother's name or "Teddy McTheoremface".
• A non-fungible token (NFT) attesting ownership of the theorem and the name you have chosen for it.
• Anyone citing this result will be required to note that you own it and use the name you chose above. You cannot, however, limit the use of the theorem or receive compensation for its use. 
• By virtue of owning this theorem you will a Fortnow number of zero. This immediately gives you an Erdős number of 2. If you have previously written a paper with Paul Erdős then both of us will now have an Erdős number of 1.

Frequently Asked Questions

Q: Why this theorem?

A: The theorem is in one of my few solely authored papers and I can't afford to share the proceeds of the sale. 

Q: Doesn't Ryan Williams and others have stronger theorems?

A: The results are incomparable. Ryan gives bounds on a single algorithm with low time and space. My theorem allows different machines for the time and space bounds.

Also, to the best of my knowledge, Ryan's theorem is not for sale.

Q: Doesn't the proof of the theorem rely on other people's theorems such as Nepomnjascii? Shouldn't he get some of the value from this sale?

A: I'm not selling the proof of the theorem, just the theorem itself.

Q: If I purchase this theorem will I get to write next year's April Fools post?

A: No.

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?