Wikipedia:Reference desk/Archives/Mathematics/2014 August 2

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August 2

Sharing birthdays

Not quite the birthday problem. Someone with a lot of Facebook friends posted that it was remarkable today that, according to the website, it wasn't the birthday of any one of their friends today. If all their friends entered their birthday on the website, how many friends would they need to have in order to have a less than 50% chance of such a day, on which nobody they know has a birthday? Warofdreams talk 01:08, 2 August 2014 (UTC)

This is OEIS: A073593 with n=365. Unfortunately the table only goes to n=59 and my spreadsheet broke when I tried to go up to 365. It appears to be a nearly linear relationship and extrapolating gives a ballpark estimate of about 2000.--RDBury (talk) 05:33, 2 August 2014 (UTC)

This makes one wonder about the kind of things OEIS will list. One can generate an integer sequence from any real function over the integers using the floor and ceiling functions. —Quondum 16:14, 3 August 2014 (UTC)

Are you sure about that? A073593 would have you possibly checking the same person N times, rather than checking N people each 1 time. --jpgordon^{::==( o )} 19:31, 2 August 2014 (UTC)

Not sure I understand the issue, N here is the number of days in the year, not the number of people. Anyway, the methodology was to compute the values by spreadsheet up to n=20 or so (i.e. what would happen if there were only 20 possible birthdays?) then search for those values in the OEIS. So this sequence matches what I got independently up to the first 20 values, which is a good empirical argument that it's the right sequence. --RDBury (talk) 02:59, 3 August 2014 (UTC)

There's a straightforward approximation if we ignore February 29 and pretend that each day is independent. Given n people, the probability that a given day has no birthday is $p=(364/365)^{n}\approx e^{-n/365}$. Assuming independence, the problem asks that $(1-p)^{365}=1/2$, which gives us another expression for p: $p\approx {\frac {\ln 2}{365}}$. Combining, we get $n\approx 365\ln {\frac {365}{\ln 2}}\approx 2287$. Egnau (talk) 04:27, 3 August 2014 (UTC)

... which is consistent with the asymptotic formula mentioned in OEIS: A073593, which is $d(\ln d+c)$ for d days in the year, with c approximately 0.36. Your approximation suggests $c=-\ln \ln 2\approx 0.366513$. Gandalf61 (talk) 16:46, 3 August 2014 (UTC)

I'm unsure whether "such a day" in the question means that today is such a day or that there exists such a day. If it's the former then we are overcomplicating the matter but I will also assume it's the latter. My brute force exact computation also says there must be at least 2287 friends to have at least 50% chance that there is no day without a birthday. My program agrees with the 59 terms in oeis:A073593. PrimeHunter (talk) 17:08, 3 August 2014 (UTC)

Rate of ageing calculation

If an extraterrestrial's 500 years is 80 years in human equivalent (assuming, for the sake of calculation, that 80 is the average human lifespan), does 500:80=6,25 give their correct rate of ageing compared to the human one? (in other words, is the phrase "the extraterrestrials live 6,25 times longer than humans" the same as "they age 6,25 slower than humans"?) What is the calculation template in such cases (I presume it's the same as in progeria cases)? --93.174.25.12 (talk) 15:36, 2 August 2014 (UTC)

It's probably not that simple. For example, lets compare humans to dogs. It's said that dogs age some 7 times faster, but that's just an average. They become adults in a year or two, so that's more like 10-20 times faster. On the other hand, they die at maybe 10-15 years, so that's more like 5-10 times faster. So, not all phases of aging are proportional. I'd expect the same with aliens. They may even have additional stages of life, beyond childhood and adolescence. Some insects have strange life cycles, for example, they may live for years as a grub, but only briefly as an adult, like periodical cicadas. StuRat (talk) 03:08, 3 August 2014 (UTC)

Ok, but how the rate of ageing of, say, children with progeria is calculated? I've read they age 10 times faster than normal (which, unless I'm not mistaken, means that each their year equals 10 normal years, which is rather too fast, I think). How that rate is calculated anyway? --93.174.25.12 (talk) 08:38, 3 August 2014 (UTC)

I have no idea. But, my guess is that they look at the average lifespan for a non-afflicted person (let's say that it's 80 years old) and the average lifespan for a child with progeria (let's say that it's 8 years old). Perhaps they get the calculation from there? In this case, 80 divided by 8 equals 10. Therefore, the "average" progeria afflicted child ages ten times faster than the "average" non-afflicted person. Just a guess on my part. But, it does seem to make sense. In other words, when "they" compare the aging rates of two separate classes of people, they are essentially saying "at what rate does that person progress from his birth to his death?" (i.e., what is his life span?). I think. Joseph A. Spadaro (talk) 01:59, 4 August 2014 (UTC)

But, again, that doesn't mean they age proportionally. They don't hit puberty by age 2, for example. StuRat (talk) 20:33, 4 August 2014 (UTC)

Yes, agreed. I imagine that "milestones" such as puberty, etc., are not reached in a proportional amount of time. Joseph A. Spadaro (talk) 03:48, 6 August 2014 (UTC)

Zonal spherical harmonics

Hi there, In the Wikipedia article on Zonal spherical harmonics one can read in the subsection on Relationship with harmonic potentials this phrase:

"The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn; to wit, for x and y unit vectors,..."

What's "to wit?" Is it a special term or it should be understood in a sense of the common language?

Thanks, --AboutFace 22 (talk) 16:51, 2 August 2014 (UTC)

It should be understood in the common English language sense of the term; that is, "that is to say...". RomanSpa (talk) 17:13, 2 August 2014 (UTC)

It is an idiom in formal writing, meaning the same thing as the informal expression "that is to say". Sławomir Biały (talk) 17:30, 2 August 2014 (UTC)

Personally, I'd change it. Math writing tends to be jargony enough without expressions you'd only read in a legal contract. --RDBury (talk) 18:46, 2 August 2014 (UTC)

I first came across it some 35 years ago in "The Owl Who Was God", a James Thurber fable. The copy I currently have is

• Thurber, James (1991) [1951]. "The Owl Who Was God". Fables for Our Time. London: Mandarin. p. 27. ISBN 0-7493-0831-1. 'Can you give me another expression for "that is to say" or "namely"?' asked the secretary bird. 'To wit,' said the owl.

That story first appeared in The New Yorker, 29 April 1939. --Redrose64 (talk) 19:16, 2 August 2014 (UTC)

Thank you, I appreciate the explanations. It widens my horizon also but I was baffled when I saw it. First I tried to look for a mathematical term, you never know. Thanks, --AboutFace 22 (talk) 20:22, 2 August 2014 (UTC)