Wikipedia:Reference desk/Archives/Mathematics/2011 December 9

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December 9

Function

I need to find a function f(k) with k = 1, 2, ... , n subject to the following:

• f(1) is known
• f needs to match the "pattern" of another known function, g(k) and
• f(n) = g(n) but f(1) ≠ g(1)

So if g starts at some number and then increases and then decreases and ends up at 20 then f needs to start somewhere else and then increase and then decrease and end up at 20. So given g and given f(1), how do I find f? Thank you. --163.202.48.108 (talk) 10:07, 9 December 2011 (UTC)

There is no thing that could, in general, be termed a "pattern" unless it were something like a turing machine, but from what you are saying, I'm guessing you something more along the lines of a recurrence relation or that you have some initial part of g and can "see" the "pattern". If you can give me some more explicit information, I'd be happy to help you work this out. 209.252.235.206 (talk) 10:20, 9 December 2011 (UTC)

So if f(1) is 50 and n = 10 and g(k) = 2k for k = 1 ... 10 then I want f to decrease linearly from 50 to 20. But in my work g is not linear. If n = 5 and g is say {10, 40, 1, 30, 100} then f will be something like {50, 70, 10, 45, 100} - I just made up those numbers in between. --163.202.48.108 (talk) 10:26, 9 December 2011 (UTC)

That's the thing, there is no thing or pattern that absolutely exists for some sequence of numbers that can be applied to some other initial value; although, you can have some rule that you can apply and get different sequences for different initial values. What I would need to know is either how you are getting the values for g in the first place or, at least, what you are attempting to do. Outside of this, there is really no general answer that can be given; really, all that can be done is to try and work out what specifically we mean by "pattern" in this case, then apply it. Phoenixia1177 (talk) 10:45, 9 December 2011 (UTC)

You could consider the following: Take a map of the plane into itself that will take the line (0, 10) to (4, 100) to the line (0, 50) to (4, 100), call this map f. Then define a new sequence as f(0, 10), f(1, 40), f(2, 1), etc. This will give you a sequence constructed from the old one. If f behaves nicely (lines to lines, or some such), then it will preserve some of the structure that is implied by looking at the triangles (k, s(k)), (k+1, s(k+1)), and the line from start to end point. Of course, like I said, this isn't really an answer, just a possibility. You could just as easily rescale the differences in the first sequence/or the quotients and apply them to the new on in a way that works. Or you could look at the slope from (k, s(k)) to (k+1, s(k+1)), then use a line with a related slope and it's intersection with x = k + 1 to get the next point; though, you would have to do some scaling to ensure that you get the same endpoint. Finally, you could just pick random higher/lower than the last term points based off of if g increases or decreases, then just force that the last term be 100. What are you using this for? Note: I haven't sat down and done of these to look at how alike the sequences would be, just a few quick ideas I came up with here at work.Phoenixia1177 (talk) 11:38, 9 December 2011 (UTC)

What if you take the weighted average of f(1) and g(k)? That it, f(k) = f(1) (n-k) / (n-1) + g(k) (k-1) / (n-1), so that the "pattern" of g(k) slowly reveals itself. Would that satisfy your application? With your first example of "f(1) = 50, n = 10, g(k) = 2k" this would yield f(k) = 50 (10-k) / 9 + 2k (k-1) / 9 = (50, 44.89, 40.22, 36, 32.22, 28.89, 26, 23.56, 21.56, 20). With your second example of "f(1) = 50, n = 5, g(k) = (10, 40, 1, 30, 100)" this would yield f(k) = (50, 47.5, 25.5, 35, 100). Alternately, you could form f(k) by applying a linearly decreasing bias to g(k) so that f(1) = g(1) and f(n) = g(n). That is, f(k) = g(k) + (f(1) - g(1)) (n-k) / (n-1). For your first example this would yield f(k) = 2k + 48 (10-k) / 9 = (50, 46.67, 43.33, 40, 36.67, 33.33, 30, 26.67, 23.33, 20) and for your second example this would yield f(k) = (50, 70, 21, 40, 100). You could also choose something other than a linear decrease to the bias, perhaps with the method and rate of decrease depending on the "pattern" you wish to preserve. -- ToE 02:39, 10 December 2011 (UTC)

I seem to to remember a technique for construction of a polynomial through a given set of points. "Pattern" is indeed an unuseful term for definitions, we have recently had a civil war (almost) on Wikipedia because someone thought it was. And in this case suppose g(x) wanders around 20 for our k known points, and f(1) is a googleplex, then almost any nice f is going to differ a lot from g.

However there are other less nice things you could do, possibly Fourier transforms. Or use teh obvious answer:

f(x) = g(x) (x!=1): f(1)=c. This is a function if g is.

Rich Farmbrough, 15:50, 12 December 2011 (UTC).

Periods

Why are these things called "periods"? Staecker (talk) 12:50, 9 December 2011 (UTC)

Perhaps because they generalize π, and the periods of elliptic functions. Bo Jacoby (talk) 22:13, 9 December 2011 (UTC).

Embedding n-dimensional polyhedra into an k-dimension integer grid.

As far as I can tell, no polygon other than a square can be embedded in a two dimensional integer grid (meaning that its vertices only have integer coordinates). I've seen the proof that the triangle can't be embedded in a 2-dimensional grid, but it can in a 3-dimensional grid (1,0,0),(0,1,0),(0,0,1). It is clear than the n-cube and n-orthoplex ("octahedron analog") can be embedded in the n-dimensional grid and the n-simplex in the n+1 dimensional grid. In addition the 24-cell is certainly embeddable in the 4-dimensional grid. the permutations of (+-1,+-1,0,0)

So for the polygons from pentagon on up, the dodecahedron, the icosahedron, 120-cell and 600-cell, is there for each of them a specific n where they can be embedded in an n-dimensional integer grid?Naraht (talk) 16:59, 9 December 2011 (UTC)

Slice a cube in half with a plane through the midpoints of its sides and you get a hexagon, with coordinates I think (0, 1, -1), (1, 0, -1), (1 ,-1, 0), (0, -1, 1), (-1, 0, 1), (-1, 1, 0), lying in the plane x + y + z = 0. I don't know if this generalises though.--JohnBlackburne^words_deeds 21:07, 9 December 2011 (UTC)

I agree on the hexagon.Naraht (talk) 21:52, 9 December 2011 (UTC)

And thinking about it some more the next one to try and find is the pentagon. If you can't find that then the dodecahedron, icosahedron, 120-cell and 600-cell are all impossible as the pentagon can just be read off them. The other ones which seem most likely are 8 and 12, as 2³ and 2 × 6. I have a suspicion that none of these are possible though: three points define a plane so you really only need three dimensions; any more are surplus. And I don't think there are any other symmetries in two and three dimensions you can use.--JohnBlackburne^words_deeds 21:59, 9 December 2011 (UTC)

Suppose you can embed a regular polygon with $n$ sides. Choose any corner and let $\mathbf {a}$ and $\mathbf {b}$ be two vectors emanating from it and pointing to the previous and next corners. By Regular_polygon#Angles, the angle they span is $\theta =(1-2/n)\pi$. By Dot_product#Geometric_interpretation, $\cos \theta ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}$ which equals ${\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {a} \cdot \mathbf {a} }}$ since $\left\|\mathbf {a} \right\|=\left\|\mathbf {b} \right\|$. So a necessary condition is that $\cos((1-2/n)\pi )$ is a rational number. In particular, a pentagon is impossible because $\cos 108^{\circ }=(1-{\sqrt {5}})/4$. 98.248.42.252 (talk) 09:36, 10 December 2011 (UTC)

I'm not sure I follow things all the way through, but which n give a rational number for $\cos((1-2/n)\pi )$?Naraht (talk) 17:30, 12 December 2011 (UTC)

n=3,4,6 (assuming n ≥ 3). This can be shown using Niven's theorem. 98.248.42.252 (talk) 05:01, 13 December 2011 (UTC)

substitution and elimination methods d-rt

Is there a website that teaches you how to use both substitution and elimination methods in a math problem where it gives you only the distance and the rate and you have to figure out the time d=rt for example two planes are travelling towards each other in at a distance of 780 km with the rates of 190 and 200 km/h starting at the same time? So far, I know that you have to use the d=rt equation but how? — Preceding unsigned comment added by 70.31.19.23 (talk) 23:55, 9 December 2011 (UTC)

I'm not sure what the methods you asked are, but are you trying to determine when the two planes would cross each other's paths? If so, I'd be more than happy to help you with the problem and how to solve one's like it:-) Phoenixia1177 (talk) 08:45, 10 December 2011 (UTC)