Wikipedia:Reference desk/Archives/Mathematics/2016 January 4

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January 4

equation for any Point Inside a Triangle

as we know that we have a vector equation to get centroid of a traingle eg centroid=a+b+c/3 if a,b,c are position vectors of three corners of triangle.How to derive an equation for defining any point inside a triangle in terms of vertex position vectors of the traingle .Kindly help 112.133.223.2 (talk)

Any point in the plane of the triangle can be expressed in the form a + λ(b - a) + μ(c - a) where a, b, and c are the position vectors of the three vertices, and λ and μ are scalar multiples. See Plane (geometry)#Describing a plane with a point and two vectors lying on it. Dbfirs 11:10, 4 January 2016 (UTC)

...and a point inside or on the boundaries of the triangle has the form ra +sb + tc where r, s, t ≥ 0 and r+s+t = 1. If you want to exclude the boundaries of the triangle, replace the first condition by r, s, t > 0. See convex combination. Gandalf61 (talk) 12:33, 4 January 2016 (UTC)

See barycentric coordinates. --Tardis (talk) 18:33, 4 January 2016 (UTC)

Newton's Method

Let $f:\mathbb {R} ^{n}\to \mathbb {R}$ be a smooth function. Let H denote the Hessian matrix of f. Assuming $\forall x\in \mathbb {R} ^{n}|Hx|\leq q<1$. (|| is the absolute value function)

1. Under these assumptions Newton's method (for multiple variables) convergent without another assmuptions?
2. Assuming also that H is positive definite, than Newton's method convergent, without another assumptions?

213.8.204.4 (talk) 12:54, 4 January 2016 (UTC)

Your assumption doesn't make sense to me. Unless H is 0, you can increase $|Hx|$ (I assume this means the norm) arbitrarily by multiplying x by a scalar. Did you mean to say that $|Hx|\leq q|x|$, that is, the norm of H is q? That would obviously not be enough on its own. -- Meni Rosenfeld (talk) 14:59, 4 January 2016 (UTC)

Umm... This really doesn't make sense...

I'd just tried to generalize the univariate case: $\forall x|f'(x)|\leq q$...

Perhaps the correct generalization is $\forall x||Hx||\leq q$, where $||\cdot ||$ denotes a norm.

So, what is the correct generalization (if any exists) that implies that Newton's method convergent in the multivariate case? 213.8.204.4 (talk) 15:28, 4 January 2016 (UTC)

I think the generalization is in the article, though in not much detail. See the section "Nonlinear systems of equations". Also, you want f:Rⁿ→Rⁿ; if f:Rⁿ→R then f=0 defines a surface (usually), not a finite set of points. --RDBury (talk) 15:40, 4 January 2016 (UTC)

So Newton's method just needs to find some point in that surface. In the univariate case we usually have a set of solutions rather than a unique solution, too. 213.8.204.4 (talk) 17:39, 4 January 2016 (UTC)

If you're looking for some point on the surface then set all but one variable to a constant. The problem is then reduced to the univariate case. The univariate case has will usually have multiple solutions, but Newton's method assumes that you're already close enough to a solution so that you can ignore the others. Even in the univariate case, Newton's method doesn't work well when there are two roots close together. --RDBury (talk) 18:58, 4 January 2016 (UTC)

Ok, so what you meant is $H(x)$. I thought you meant $H\cdot x$.

So yes, you can't talk about the absolute value of a matrix, but you can talk about its norm for example. -- Meni Rosenfeld (talk) 19:09, 4 January 2016 (UTC)

Thanx :) 213.8.204.4 (talk) 10:23, 5 January 2016 (UTC)

An elementary question about bookies

What does it mean that the odds of someone winning an election is 8/11 or -1000? (You find both ratios and signed integers at different bookmakers, but the first format seems to be much more prevalent) How do you make sense of that and how do you convert one format into the other? Contact Basemetal here 16:07, 4 January 2016 (UTC) PS: Is this the right desk for this question?

See http://www.olbg.com/school/lesson10.htm

8/11 means if you bet £11 and win you will get your stake (£11) back plus winnings of £8 (apart from a taxes that may be due).

+1000 means that if you bet £100 and win, you will get £1000 plus your £100 stake (less taxes). (What you win for a 100 bet)

-1000 means if you bet £1000 and win, you will get £100 plus your £1000 stake (less taxes). (what you bet for a 100 win)

-- SGBailey (talk) 17:18, 4 January 2016 (UTC)

Great. Thanks. Contact Basemetal here 18:06, 4 January 2016 (UTC)

Wikipedia article: odds. Joseph A. Spadaro (talk) 18:32, 4 January 2016 (UTC)

I've taken a table from odds and copied it to my sandbox User:SGBailey/Sandbox where I have edited it. Could some of you have a look at the table and let me know if it is (a) correct and (b) if you think it would be useful to replace the 'odds' table with the sandbox expanded version. -- SGBailey (talk) 11:42, 5 January 2016 (UTC)

My confusion with this (mostly with the article) is that the -100 and +100 are listed as "US Odds". I've never heard them used here in the US but am very familiary with odds looking like 3-1 or 9-2 for standard horse races. So identifying those as US Odds seems wrong.Naraht (talk) 17:26, 6 January 2016 (UTC)

The odds article lists them (+1000 and -1000) as "Moneyline odds favoured by Americican bookies." I took that to mean US odds, but apparently not. -- SGBailey (talk) 23:58, 6 January 2016 (UTC)

I've never heard them used here in the UK, so if they are not American, where are they used? Dbfirs 17:28, 6 January 2016 (UTC)

And remember, gambling is unethical and immoral...unless the politicians get a cut, in which case it should be encouraged and advertised, at least according to them. StuRat (talk) 20:43, 6 January 2016 (UTC)

I've always viewed state run lotteries to be extra taxes on the mathematically uneducated.Naraht (talk) 21:19, 6 January 2016 (UTC)

Hacettepe Journal of Mathematics and Statistics - notability

New journal page. Is this notable for journals? I don't know enough about maths journals to say. Blythwood (talk) 17:19, 4 January 2016 (UTC)

Just so the question doesn't go without a response, this is the journal of the math department at Hacettepe University, located in Turkey. So maybe it's the Turkish equivalent of the Illinois Journal of Mathematics. Not sure what that means in terms of notability, though it might be worth mentioning that the IJM does not have its own WP article. --RDBury (talk) 04:45, 6 January 2016 (UTC)