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January 9

Function for a catenary given unequal heights

All treatments of catenaries involving unequal heights (h1 and h2) seem to be only interested in arc length. What is the general function for any catenary connecting two unequal heights, y = f(x)? Arc length is a secondary concern. — Preceding unsigned comment added by 98.14.205.209 (talk) 00:55, 9 January 2018 (UTC)

The general formula for a catenary, allowing for scaling and translation, is

$y-k=a\cosh \left({\frac {x-h}{a}}\right).$

There are three parameters so requiring that the curve pass through two points does not allow you to solve for all three and get a specific equation. You need some additional piece of information, such as the arc length, to determine the curve. It might be possible to determine the catenary passing through three given points, but I don't know how complicated the expressions get. --RDBury (talk) 02:49, 9 January 2018 (UTC)

You have one degree of freedom in your problem, so, as RDBury said above, you need additional condition to solve for one free parameter.
Given a horizontal and vertical displacement you have a slope of a line segment, which shall become a chord of a catenary arc. You can have multiple chords with the same slope on a single catenary curve. Each of them can be scaled to your original problem thus giving different catenary arcs through the two given points. See images in Catenary # Mathematical description # Equation section.
So you can choose some additional constraint (say, the arc length, the maximum curvature, the height of minimum point etc.) to make a solution unique (which does not necessarily mean easy). --CiaPan (talk) 10:35, 10 January 2018 (UTC)

January 12

Life expectancy when you are a baby vs. when you are old

If you are just born into a society where your life expectancy is 80, then someone could say "...your life expectancy is 80..." right?

What if you're, say, 75? You've already survived childhood illness and car crashes in your '20s, right?

So, could someone say "...your life expectancy is now 85..."?*

• (All things being equal.)

Anna Frodesiak (talk) 01:40, 12 January 2018 (UTC)

The third paragraph of our article Life expectancy states, "Mathematically, life expectancy is the mean number of years of life remaining at a given age, assuming age-specific mortality rates remain at their most recently measured levels." According to that definition, your octogenarian has a life expectancy of 5 years, not 85.

Check out the last paragraph of Life expectancy#Life expectancy vs. life span, and that paragraph's last reference, Life Expectancy by Age, 1850–2011. A U.S. white male (the first table) born in 2000 had a Life Expectancy at Birth (LEB) of 74.8 years, while an 80-year-old in 2000 had a life expectancy of 7.6 years, though he had an LEB of only 56.34 years (back in 1920, when born). -- ToE 02:53, 12 January 2018 (UTC)

Thank you, ToE. Sorry about how I worded my question. Of course, I meant remaining life.

If that person you refer to had an LEB of only 56.34 years back in 1920, that was partially because they didn't anticipate a future improvement in healthcare etc., and partially because childhood disease then was more of a consideration, right?

My question would be about a baby born today and an 80-year-old today, assuming that nothing on Earth changes in terms of healthcare improvements, etc.

So, is the main consideration about the bullets the 80-year-old dodged in the past, and what ones are upcoming? And my guess about that (probably wrong) is that child mortality and wars and such are a lot lower now that before, so the 80-year-old is on track to drop dead within a couple of years if the expected lifespan is 80ish. Is that the way to reckon it? Cheers, Anna Frodesiak (talk) 03:10, 12 January 2018 (UTC)

Yes, comparing the octogenarian's future life expectancy of 7.6 years to his LEB of 56.34 years isn't really fair as by definition the values come from age-specific mortality rates at the time of prediction, not taking into account healthcare improvement. Still, back in 1920, an 80-year-old (white male; first table) has a further life expectancy of 5.47 years.

So as I understand the way you expressed your main consideration, then no, the 80-year-old today is expected to live longer than an 80-year-old back in 1920. But I think I know what you are getting at.

For a given year, look at the Life Expectancy at Birth. Then look at someone who is already at that age and see what their life expectancy is. And then compare those figures for different years. For instance, back in 1920 the LEB was 56.34, and a 56.34-year-old had a further life expectancy of between 22.22 and 15.25 years, which we could interpolate to 17.80 years, for an expected age at death of 74.14. But in 2000 when the LEB was 74.8, a 74.8-year-old had a further life expectancy of between 13.0 and 7.6, which interpolates to 10.4 years, for an expected age at death of 85.2.

So in 1920 an LEB=56.34-year-old could expect to live another 17.80 years, while in 2000 an LEB=74.8-year-old could only expect to live another 10.4 years.

Is that what you were getting at? -- ToE 03:59, 12 January 2018 (UTC)

Yes, totally. Thank you, ToE. I'm really glad I understand this now. I'd bet this doesn't occur to a lot of people who are 80 in a place where that is the expected lifespan. They probably think their time will be up any second. Thank you so much for helping me to understand this. You are very kind. (I'm thinking of England too because that is where I'm from originally.) :) Anna Frodesiak (talk) 08:13, 12 January 2018 (UTC)

Someone who's 80, and who thinks their time is up any second because the average life expectancy is 80, is committing the Ecological_fallacy of thinking that an average (life expectancy) applies to an individual. I mention this because one of my statistics classes used that exact scenario to illustrate the fallacy (the example I always gave was "No family actually has 1.5 children") OldTimeNESter (talk) 00:13, 13 January 2018 (UTC)

Thank you, OldTimeNESter, for the fine input. I'll read that. Anna Frodesiak (talk) 00:32, 13 January 2018 (UTC)

You're quite welcome. Here's a (slightly) risque example of the same principle: https://www.facebook.com/TheEconomist/posts/10155060566504060. OldTimeNESter (talk) 02:32, 13 January 2018 (UTC)

I'm afraid I can't read that, OldTimeNESter. My location prohibits it. Anna Frodesiak (talk) 04:59, 13 January 2018 (UTC)

Quote: 'He uses the phrase “on average, humans have one testicle” to make the point that the mean can be a misleading description of a population'. Double sharp (talk) 11:48, 13 January 2018 (UTC)

I'm head of state of 0.00000003 countries and have died 0.9 times. Sagittarian Milky Way (talk) 18:20, 13 January 2018 (UTC)

That Facebook page merely links (with a quick summary) to this article: [1] —Tamfang (talk) 18:41, 13 January 2018 (UTC)

Thank you, Tamfang. I'll read that. Anna Frodesiak (talk) 21:14, 13 January 2018 (UTC)

The probability that the dying age X of a random person is x, is

$P(X=x)$

The expected dying age of this person is

$\sum _{x=0}^{\infty }xP(X=x)$

The conditional probability that the dying age X of a random person is x, when he is not yet dead at age 75, is

$P(X=x|x>75)$

The expected dying age of this person is

$\sum _{x=76}^{\infty }xP(X=x|x>75)$

Bo Jacoby (talk) 19:04, 14 January 2018 (UTC).

It's also interesting to consider the time dependent probability per unit time $P(x,t)$ of someone aged $x$ at time $t$ to die right at that time. If $P(x,t)$decreases sufficiently rapidly as a function of time, then the expected age of dying can be infinite. So, even though at no time the chance of dying becomes exactly zero at any age, it's still possible that you can expect to live till eternity. Count Iblis (talk) 19:34, 14 January 2018 (UTC)

January 13

natural irrational numbers

Much ado is made over the natural numbers. It seems like a similar concept should exist for numbers like $2{\sqrt {3}}+3{\sqrt[{3}]{2}}$. AFAICT you should have a mathematical group for addition, subtraction, and multiplication for numbers "${\sqrt[{n}]{x_{1},x_{2}...x_{n}}}$" = the sum of ${\sqrt[{n}]{x_{i}}}$ for positive $x_{i}$ and $-{\sqrt[{n}]{-x_{i}}}$ for negative $x_{i}$. Operations using different n should return numbers with the least common multiple of the n's, at most. I didn't immediately find something like this by searching, but do such/similar numbers have a name and any interesting properties? Wnt (talk) 07:55, 13 January 2018 (UTC)

I think you are interested in algebraic numbers, and maybe also algebraic integers for their arithmetic properties. In fact, starting from positive integers and constructing new numbers performing the four operations and extractions of n-th roots do produce all algebraic numbers (this is the Ruffini-Abel theorem). Actually there are several collections of numbers as you are describing, named algebraic fields, and their properties and characterization is the object of the algebraic number theory.--pma 09:27, 13 January 2018 (UTC)

The Ruffini-Abel theorem shows there are algebraic numbers which can't be produced using the four aritthmetic operations and roots on the natural numbers. For instance the solutions of x^5+x+1=0 are algebraic but can't be produced that way. Dmcq (talk) 14:13, 13 January 2018 (UTC)

$2{\sqrt {3}}+3{\sqrt[{3}]{2}}$ is a closed-form algebraic expression, and hence is an algebraic solution (also called a solution in radicals). Algebraic solutions are a subset of the algebraic numbers, the latter of which, as Dmcq says, also include solutions of algebraic equations that, by the Abel-Ruffini theorem, cannot be in your form. Loraof (talk) 19:22, 13 January 2018 (UTC)

See also Radical extension. Loraof (talk) 19:32, 13 January 2018 (UTC)

I have to admit that so far I remain confused, probably due to ignorance. Algebraic numbers seem to include oddities like ${\sqrt[{3}]{{\sqrt {2}}+1}}$ that aren't in the set I had in mind... well, then again I don't know that, but I wouldn't think so. Also, I personally have no idea whether any polynomial has to exist that has a solution ${\sqrt {2}}+{\sqrt[{3}]{3}}+{\sqrt[{5}]{5}}$ , for example. And algebraic fields seem to be about linear combinations, whereas the numbers I have in mind here consist only of the presence or absence of each integer value, with no integer and its inverse appearing in the same expression. For example, the last one I mentioned is equal to ${\sqrt[{30}]{32768}}+{\sqrt[{30}]{59049}}+{\sqrt[{30}]{15625}}$ and thus might be written "${\sqrt[{30}]{15625,32768,59049}}$" The numbers have to be unique in "reduced form" because if any two match up you just multiply by 2^root and replace them with that. It seems simple enough and pretty enough I feel like it must be old hat somewhere. Wnt (talk) 22:44, 13 January 2018 (UTC)

According to Wolfram Alpha, the minimal polynomial of ${\sqrt {2}}+{\sqrt[{3}]{3}}+{\sqrt[{5}]{5}}$ has degree 30. -- ToE 03:58, 14 January 2018 (UTC)

More generally, an expression of the form ${\sqrt[{n}]{x_{1}}}+{\sqrt[{n}]{x_{2}}}+\dots +{\sqrt[{n}]{x_{k}}}$ will satisfy an equation of degree n^k, and so will be algebraic. The notation you have, ${\sqrt[{30}]{15625,32768,59049}}$ to mean ${\sqrt[{30}]{15625}}+{\sqrt[{30}]{32768}}+{\sqrt[{30}]{59049}}$, seems interesting, and you seem to asking whether there is a use for the set of numbers to which it applies. I won't claim the answer is a definite no, but historically it seems more useful to study either more specific forms (as in quadratic fields) or more general forms (as in Galois theory). Not that there's anything wrong with trying to find out interesting things about a middle ground, but math is not always cooperative when it comes to actually doing it. To start with, you might look at the Kronecker–Weber theorem. --RDBury (talk) 05:11, 14 January 2018 (UTC)

I think the minimal polynomial article implies every one of these numbers does solve a polynomial, because you should just be able to set up a bunch of factors $(x-{\sqrt[{30}]{15625}}-{\sqrt[{30}]{32768}}-{\sqrt[{30}]{59049}})(x+{\sqrt[{30}]{15625}}-{\sqrt[{30}]{32768}}-{\sqrt[{30}]{59049}})(etc.)$. For every product where you multiply an x (for example) by the positive square root there's another where you multiply the other x by the negative square root, so the radical terms all ought to cancel out leaving a polynomial with (I think) integer coefficients. Though I'll admit, I didn't actually check that in this case let alone prove it. Wnt (talk) 02:33, 15 January 2018 (UTC)

$\prod (x\pm {\sqrt[{30}]{15625}}\pm {\sqrt[{30}]{32768}}\pm {\sqrt[{30}]{59049}})$ won't work. Not only will there be 8th power constant terms you don't want (e.g. 15625^8/30), but the middle terms don't cancel out as fully as you might wish. For instance, $\prod (x\pm a\pm b)=x^{4}-2a^{2}x^{2}-2b^{2}x^{2}-2a^{2}b^{2}+a^{4}+b^{4}$. More generally, this construction will lead to those terms containing any odd powers (of variable or constant) cancelling out, but you will be left with terms containing only even powers. (Drawing two marbles from a sack containing an equal number of red and blue marbles will more likely yield a mixed pair than a matched pair.) Minimal polynomial (field theory)#Examples states:

If α = √2 + √3, then the minimal polynomial in Q[x] is a(x) = x⁴ − 10x² + 1 = (x − √2 − √3)(x + √2 − √3)(x − √2 + √3)(x + √2 + √3).

The minimal polynomial in Q[x] of the sum of the square roots of the first n prime numbers is constructed analogously, and is called a Swinnerton-Dyer polynomial.

The key for this method is the square roots which are taken care of by the even powers. -- ToE 14:59, 15 January 2018 (UTC)

Abstractly, one can consider the tower of fields obtained by adjoining separately each of ${\sqrt {2}},{\sqrt[{3}]{3}},{\sqrt[{5}]{5}}$. Because 2, 3, and 5 are coprime, and it can be shown that the two ways of constructing the field extension are the same, one obtains the degree of the overall extension generated by adjoining ${\sqrt {2}}+{\sqrt[{3}]{3}}+{\sqrt[{5}]{5}}$ as the product of the degrees of the individual extensions (2, 3, and 5).--Jasper Deng (talk) 08:30, 15 January 2018 (UTC)

Let $\omega$ be a primitive 30^rd root of unity, say $\omega =\cos(2\pi /30)+i\sin(2\pi /30)$ . Consider the polynomial $\prod _{l=0}^{29}\prod _{m=0}^{29}\prod _{n=0}^{29}(x-\omega ^{l}{\sqrt[{30}]{15625}}-\omega ^{m}{\sqrt[{30}]{32768}}-\omega ^{n}{\sqrt[{30}]{59049}})$. Bo Jacoby (talk) 23:11, 15 January 2018 (UTC).

January 14

Calendar

Week before and after the 30th day of a month. Please specify the correct dates please. 37.111.234.166 (talk) 16:28, 14 January 2018 (UTC)

Your question is unclear. --70.29.13.251 (talk) 20:52, 14 January 2018 (UTC)

Target the 30th date of one month. what date is the week before and the week after. My calculation is 16th or 23rd (before) and 6th or 13th. 119.30.38.208 (talk) 16:36, 15 January 2018 (UTC)

You can analytically continue each month in terms of the next and previous months. In case of August you define the 32nd of August to be the 1st of September and in general x August = (x - 31) September and similarly for other month. Count Iblis (talk) 16:43, 15 January 2018 (UTC)

Product price mark-up

A product cost £10 to buy. It consist of the all the investments implemented before customer received at hand after purchase.

The tax percentage is, say for example 20%.

1) Do you tax 20% of the £10, or do you exclude the investment (say £5) and then tax the profit sum?

37.111.234.166 (talk) 16:30, 14 January 2018 (UTC)

See Value-added tax for the European system. Dbfirs 21:35, 14 January 2018 (UTC)

Chart ideas for The World's Billionaires

Might there be a nice chart layout for The World's Billionaires that shows years and people and new heights? If you can think of a clever presentation, please suggest it at Talk:The World's Billionaires#Table showing top 5 or something.

Many thanks. Anna Frodesiak (talk) 21:32, 14 January 2018 (UTC)