Wikipedia:Reference desk/Mathematics

February 2

English

What is noun phrase — Preceding unsigned comment added by Ibrahim shaaibu (talk • contribs) 01:30, 2 February 2022 (UTC)

This is really a question for the Language section, but see our Noun phrase article. All noun phrases in the following lines are underlined:

Twinkle, twinkle, little star,

How I wonder what you are!

Up above the world so high,

Like a diamond in the sky.

The noun phrase in the last line contains two embedded noun phrases: a diamond and the sky.  --Lambiam 02:44, 2 February 2022 (UTC)

February 3

Definition of a topological space by neighborhoods

In the article on "topological space" the section Definition by Neighborhoods lists four axioms. I don't see how axiom 4 is independent.

Axiom 1 says: If N is a neighborhood of point x, then x is an element of N.

Axiom 4 says: If N is a neighborhood of point x, then N contains a neighborhood M of x such that N is a neighborhood of every point in M.

Doesn't 4 follow directly from 1? The singleton set containing only the point x is a subset of every neighborhood of x, so it always satisfies the conditions for neighborhood M in 4.

What am I missing here? Can you point me to a description of axiom 4 that I can understand without a deep background in topology? — Preceding unsigned comment added by Bronco creek (talk • contribs) 19:01, 3 February 2022 (UTC)

What you're missing is that the singleton {x} might not be a neighborhood of x. --Trovatore (talk) 19:09, 3 February 2022 (UTC)

Thanks. That'll teach me to trust my intuition. — Preceding unsigned comment added by Bronco creek (talk • contribs) 19:54, 3 February 2022 (UTC)

The key is not so much not to trust intuition, as to get a more trustworthy intuition.

Intuition is fundamental to mathematics. You simply can't do anything nontrivial without it. But you have to develop it. And you have to keep testing it, looking for places where it doesn't serve you well, and be willing to change those.

In this particular case, the intuition that might have served you is that a neighborhood is a set that contains a given point, plus a little margin around it. That ordinarily rules out singletons. Note that it doesn't always rule out singletons, but that's the base case you can have in mind, being aware of exceptions. --Trovatore (talk) 20:19, 3 February 2022 (UTC)

The key is not so much not to trust intuition, as to get a more trustworthy intuition. Words to live by. --RDBury (talk) 22:58, 3 February 2022 (UTC)

February 7

Height of Simplexes

I'm working my way through Simplex and have been unable to find exactly what the height of each simplex is. I'm defining the "height" of an n-simplex as the distance from one vertex to the center of the n-1 simplex made up of the other vertices. So the "Height" of a 2-simplex is sqrt(3)/2. and the "Height" of a 3-simplex is sqrt(6)/3 (according to Tetrahedron). I'm presuming that they tend to 0 as n goes to infinity, but I don't know if there is an easy formula.Naraht (talk) 09:37, 7 February 2022 (UTC)

That obviously depends on the choice of a vertex, as in 2D example you may have three different heights of a triangle, depending on a vertex chosen as an apex. --CiaPan (talk) 10:21, 7 February 2022 (UTC)

I think we are assuming that it is the "standard simplex", which will be symmetric with respect to choice of vertex. See a discussion here: https://math.stackexchange.com/questions/1697870/height-of-n-simplex showing that this height is ${\sqrt {\frac {n+1}{2n}}}$. Your answer is different I guess because you assumed the simplex edge length is 1? The typical construction of the standard simplex makes the edge length ${\sqrt {2}}$. Staecker (talk) 12:34, 7 February 2022 (UTC)

The formula ${\sqrt {\frac {n+1}{2n}}}$ works for edge length $1$; the height of a 1-dimensional simplex is the length of its single edge, so just put $n=1$ and indeed $1$ pops out.  --Lambiam 13:07, 7 February 2022 (UTC)

Yes Lambiam is right- I popped it out wring! Staecker (talk) 14:39, 7 February 2022 (UTC)

So sqrts of 2/2, 3/4 , 4/6, 5/8, 6/10 , so 1, sqrt(3)/2, sqrt(4/6)= sqrt(6/9) = sqrt(6)/3, sqrt(5/8) = sqrt(10)/4, sqrt(6/10)=sqrt(3/5)=sqrt(15)/5 etc. which as n-> inf approaches sqrt(1/2) = sqrt(2)/2. Interesting.

In contrast, the volume approaches $0$ more than exponentially fast; see the last paragraph of Simplex § Volume. The volume formula gives an alternative way to derive the height formula, using an obvious way of obtaining the volume of the regular $(n{+}1)$-simplex from that of its predecessor by integration over the height.  --Lambiam 21:10, 7 February 2022 (UTC)

February 8

Multiplication methods

2 310(35) — Preceding unsigned comment added by 41.114.170.234 (talk) 19:51, 8 February 2022 (UTC)

Do you have a question for us? You only posted some numbers. --Jayron32 20:28, 8 February 2022 (UTC)

2310 = 35 × 66. Is that somehow related?  --Lambiam 21:16, 8 February 2022 (UTC)