## Archive for the ‘Mathematics’ Category

### Slutty T-Rex

November 30, 2023

đ đ tiger tiger for ultimate November, also St. Andrew’s Day (Scotland’s national day); meanwhile, I bring you two dinosaurs trading ideas about popularity and sluttiness

A pair of Ryan North’s Dinosaur Comics strips, coming in succession on 11/10 and 11/13, in which T-Rex rambles on to his buddy Utahraptor about a fairly well-known paradoxical-sounding phenomenon in social networks, the friendship paradox. Actually, it applies more generally, and I’ll talk you through the (apparent) paradox in the general case. Yes, I’ll get to the comics, and to the way T-Rex uses the adjective slutty, but first let’s talk about your lunch partners.

The symmetric-relation paradox. Brace yourself for some mathematician-talk, but don’t despair: I’ll work up a concrete example (about you and your lunch partners) along the way.

Consider a a set N (for example, the set of people in a social network) and a symmetric relation R between members of N; R might be being friends with, say, or having gone to grade school with or — my concrete example —Â having had lunch with. Then for any member m of N (like you, for definiteness), define m’sÂ R:N-cohortÂ to be the set of members of N that m bears R to (like, the set of all your lunch partners), and m’sÂ R:N-indexÂ to be the size of m’s R:N-cohort (like, how many lunch partners you’d had). Then it can be shown that, on average, the R:N-indices of members of m’s R:N-cohort are greater than m’s R:N-index — like, on average, the number of lunch partners your lunch partners have had is greater than the number of lunch partners you have had. Yes, it sounds paradoxical. But it’s provably so.

Now, listen up: what the symmetric-relation paradox does not say is that (all) your lunch partners have more lunch partners than you do. That would be genuinely paradoxical. All it says is that the (arithmetic) mean of their lunch-partner figures is higher than yours, which is a great deal less thrilling (though it still has a whiff of the perverse about it). So let’s look at the special case, the friendship paradox,Â where N is a social network and R is the being friends with relation (which is where T-Rex starts in his Dinosaur Comics ramble, before he goes on to the having had sex with relation (parallel to the having had lunch with relation) and to sluttiness, having had many sexual (rather than lunch) partners.

### A letter from an old friend

July 27, 2023

Among the letters from the 7/31 issue of the New Yorker, under “Cultural Studies”, this erudite letter from Stephen Isard of Philadelphia PA, about Peter Hessler’s piece “A Double Edcation” in the 7/3 issue:

One of the math problems that Hesslerâs daughters attempt to solve, as part of their challenging Chinese curriculum, asks them to find the smallest number that leaves the remainders 2, 3, and 4 when divided respectively by 3, 4, and 5. Is this a trick question of the sort that Hessler depicts his children completing in third-grade math class, the kind designed to trip students up? No. What he describes is a simple introduction to a celebrated mathematical theorem known in the English-speaking literature as the Chinese remainder theorem, which guarantees that any such problem has a solution, so long as none of the divisors (in this case, 3, 4, and 5) have a factor in common other than 1. The theorem has been attributed to the Chinese mathematical text âSunzi Suanjing,â which was completed between the third and fifth centuries A.D., and it plays an important role in Kurt GĂ¶delâs proof of his incompleteness theorem. Applied here, it gives the answer to the twinsâ problem as 59.

Wow, Steve Isard is one of my oldest friends, going back to Cambridge MA in the early 1960s. We collaborated on some little papers in mathematics then, and eventually Steve and his wife Phoebe Acheson Isard (long gone, alas) became close friends with me and my wife Ann Daingerfield Zwicky (also long gone, alas). What a delight to see him still in the education business!

### From the annals of naming: twins

April 28, 2023

(This is very much a not-dead-yet posting. I’m just barely hanging on through still more medical afflictions, but now think the little time I’m able to devote to posting is better spent on upbeat things rather than intimations of mortality.)

Discovered on my desktop, a three-part Saturday Morning Breakfast Cereal from 12/19/18. Part 1, the teaser:

(#1)

### The poutine checkup

March 17, 2023

Yesterday’s (3/16) Wayno / Piraro Bizarro, with poutine checkup as an outrageous pun on routine checkup — something for the celebratory days of mid-March, from Pi Day on the 14th through St Patrick’s Day today:

(#1) A regular checkup on the state of the patient’s poutine — the fries, cheese curds, and gravy of this Quebecois specialty (If youâre puzzled by the odd symbols in the cartoon â Dan Piraro says there are 7 in this strip â see this Page)

Meanwhile, the patient is presented as stereotypically Canuck — among other things, wearing a tuque (that knitted wool watch cap) and earmuffs.

Wayno’s title for the cartoon:Â Say “eh” — evoking another piece of Canadianness, the discourse particle eh; see, from myÂ 5/18/14 posting “… plus four”, thisÂ New Yorker cartoon by Benjamin Schwartz:

(#2) OnÂ the (mostly) Canadian discourse particle eh, used (roughly) to confirm the attention of the listener (but appearing here in a medical exam in the place of ah); Canadianness is signaled by the RCMP uniform

### For Alan Turing

June 23, 2020

On the occasion of Alan Turing’s birthday today, this release from NPL:

1912 â 1954:Â  Alan Turing’s work was instrumental in placing NPL at the forefront of computer technology.

Turing had already achieved a great deal before he started work at NPL. While at King’s College, Cambridge, he earned a scholarship, Maths Tripos Part II Distinction, fellowship and Smith’s Prize, as well as writing his paper onÂ Computable Numbers. He then moved on to Princeton University and earned his PhD in 1938, before moving back to Cambridge and starting work at the Government Code and Cryptography School in 1939, where he was an essential part of the work to break the German Enigma code.

After the war he moved to NPL in 1945, and produced his plans for the ACE computer in 1946. He worked at NPL on the ACE until he left (after being on leave to Cambridge) in 1948, not long after writing hisÂ Intelligent MachineryÂ paper.

Two things here: the identity of NPL; and more on celebrations of Alan Turing.

March 4, 2020

The briefest of celebratory woofs for the 16th (= 10h-th) birthday of Opal Eleanor Armstrong Zwicky, today:

(#1)

(#2)

You are young, Opal Eleanor, the old man said,
And your hair colors wander astray,
You’ll be 91 starting today.

(#3)

### 79

August 29, 2019

As we slide into a US holiday weekend — leading to Labor Day, the first Monday in September, this year on the 2nd — my birthday (on the 6th) looms as well. Coming up is a prime-th birthday, the 79th, an auspicious number to my mind, just one short of the 80th, which many view (like the similarly vigesimal 20th, 40th, and 60th) as a landmark birthday, in this case the gateway into old age. But for the moment I’m prime, baby.

### Golden Meanies

July 12, 2018

On Sandra Boynton‘s Facebook page yesterday:

Today is Pet Photo Day, so here is a recent snapshot of my semi-domesticated Golden Meanie, Fibonacci.

(#1)

Golden Meanie is a bit of complex language play, combining the mathematical termÂ golden mean (akaÂ golden ratio); a reference to the Blue Meanies of the animated film Yellow Submarine; and a reference to the amiable domestic dog breeds the golden retriever and the golden Labrador. Plus, the name of Boynton’s Golden Meanie, Fibonacci, is a reference to the Fibonacci sequence in mathematics, which is intimately related to the golden ratio.

### Pi Day + 1

March 15, 2018

A late entry in the Pi Day sweepstakes: a Scott Hilburn cartoon from 2013:

Today’s the Ides of March and also Penis Festival in Japan, but neither of these fits the cartoon.

### Pi Day 2018

March 14, 2018

March 14 — 3.14 — is Pi Day, a day to celebrate the irrational number đ. The usual pun for Pi Day is as Pie Day — restaurants have specials on pies — or some play on the adjective irrational — but the Mental Floss site has gone further afield:

(#1)