Math: Fleeting Thoughts

[FancyHat]

I think you're talking about Euclid: The Game.

Thank you for posting that; that was fun!

[somehow]

You're very welcome! It is indeed a lot of fun.

[bachaddict]

Can you solve this problem?
http://www.futilitycloset.com/2015/01/1 ... s-problem/


A particle starts at rest and moves with velocity along a 10-foot ladder, which leans against a trough with a triangular cross-section two feet wide and one foot high. Sand is flowing out of the trough at a constant rate of two cubic feet per hour, forming a conical pile in the middle of a sandbox which has been formed by cutting a square of side x from each corner of an 8″ by 15″ piece of cardboard and folding up the sides. An observer watches the particle from a lighthouse one mile off shore, peering through a window shaped like a rectangle surmounted by a semicircle.

(a) How fast is the tip of the shadow moving?
(b) Find the volume of the solid generated when the trough is rotated about the y-axis.
(c) Justify your answer.
(d) Using the information found in parts (a), (b), and (c) sketch the curve on a pair of coordinate axes.

[Dopefish]

Oh man, I think I probably could if I really wanted to, but that's the sort of thing my physics comrades and I would make up as example of stuff the profs could put on tests that would just make us go "Nope. Guess I'm failing, thats way too much mathematical muck to trudge through." despite simultaneously being 'fair' in the sense we had the tools to solve it.


I also appreciate that Euclid the game was posted. My formal education never had me doing constructions of that nature, so I'm glad for the opportunity to try my hand at that sort of thinking. I haven't seriously sat down with it yet, but I did the first half dozen levels in the minimum number of moves by some miracle, despite none of the constructions being obvious to me at first glance, which is a nice place for the difficulty to be.

[Carlington]

I'm at roughly the same point as you, Dopefish, in terms of my knowledge relative to the game's difficulty curve. I used it as a sort of learning tool, though, in that I went and taught myself about the concepts I didn't understand. I've not quite finished it.

[doogly]

It didn't recognize one of my solutions, which made me mad.

[FancyHat]

It didn't recognize one of my solutions, which made me mad.

Which one?

[doogly]

level 12 - construct a given angle over here on this other line.

[FancyHat]

level 12 - construct a given angle over here on this other line.

I solved it in four (using all tools), but had to read the comments to learn how to do it in three. It seems to be a hard one.

What did you try?

[doogly]

Yeah much better. I was devolving things into a slide and rotation, so my 8 moves is far from "optimal," but I like it.

[Carlington]

I've been teaching myself some calculus out of the book that can be found in PDF format here. Working through the section on multivariable calculus now, and I was going along at a nice pace until I hit the section on variable changes for integrating functions of two or more variables. In particular, I understand how to work out the Jacobian and how to use that to find ways to substitute variables and get a function that's easier to integrate. I also understand how to transform from rectangular coordinates to polar coordinates (and I can do the same for spherical and cylindrical coordinates in R³ if pressed, though I'd rather not have to do spherical ones if it can be avoided.)
What I'm stuck on at the moment is the following problem, which I'm sure once I understand it I'll know what's going on, but I can't even understand what I'm failing to understand:
"Find the volume of the solid inside both x² + y² + z² = 4 and x² + y² = 1".
I know I'll need to integrate two or three times but I can't figure out what to integrate, or over what region, and it's driving me up the wall.

EDIT: Wait, I realised my problem, wasted post. What I thought was a circle was actually a cylinder. I didn't realise that it was a function of x y and z.

[Eebster the Great]

Well x² + y² = 1 is a circle in the xy-plane. If it's intended to represent a cylinder, you need a max and min z.

[doogly]

You don't, the sphere provides the limits in that direction.

[Eebster the Great]

Oh I misread his post. I thought he was finding two separate volumes, not the volume of the intersection.

[Dopefish]

I feel like I should admit up until reading the wiki page for discrete math a moment ago, I'd been mentally selling 'discrete mathematics' short by thinking of it as basically calculus but without the limit taking (i.e. 'easier' but a bit messier). Sort a middle ground of mathematical maturity between 'math for liberal arts' and 'math for math majors' for computer scientists in a way.

It's got graph theory and logic and combinatorics and number theory and a bunch of other meaty stuff to it though, so my apologies to any comp-sci people whose mathematical abilities I may have been condescending towards at any point in my life because they 'only' took discrete.

[doogly]

No that's accurate. Combinatorics, number theory and graph theory as subjects are serious, but "discrete math" as a course is essentially as your described it.

[Dopefish]

Ok, well then I guess my apology is more directed towards the overall field of discrete math as a subject, rather than the course and/or students taking such a course.

I think my brother might have a discrete math textbook lieing around somewhere, I should probably just take a look to see how trivial (or not) it seems.

[ahammel]

Is there a name for the associative binary operation in a semigroup, or is it just called an associative binary operation?

[Eomund]

Is there a name for the associative binary operation in a semigroup, or is it just called an associative binary operation?

According to wikipedia,

The binary operation of a semigroup is most often denoted multiplicatively: x·y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x,y).

[ahammel]

That's a symbol, not a name.

[jestingrabbit]

I'd call it the semigroup operation or multiplication. That's pretty unambiguous.

[notzeb]

You could probably also get away with calling it composition (in analogy with the semigroup of functions from a set to itself).

[MostlyHarmless]

The maximum possible temperature of a thermodynamic system is the limit-supremum as energy goes to infinity of energy divided by volume entropy. If you've got tex:

T_{max} = \limsup_{E\to\infty}\frac{E}{\ln V(E)}.

Maybe that sounds like it should go in the science forum, but it mostly comes from some nice properties of Laplace transforms. (Incidentally, I've been reading David Widder's book The Laplace Transform, and it's fascinating.)

[cyanyoshi]

Consider an N-sided regular polygon of side length one. Now connect each pair of vertices with a line segment, like so for N=6:

What do you think the sum of the lengths of the line segments is?

Spoiler:

I found it to be N/4*csc²(pi/(2N)) for all N ≥ 2.

First of all, I was shocked how simple the end result was. Second of all, this formula gives a strange result for N=1. I guess that makes sense, since you run into trouble trying to define angles within a 1-gon. Doubly so if you try to impose that unity side length requirement. So to help fix this, I set up a similar question:

Consider N points equally spaced around a circle with radius one, and connect every pair of vertices with a line segment. What is the sum of the lengths of the line segments?

Spoiler:

N*cot(pi/(2N))

How nice.

[jestingrabbit]

Recent meanderings found this

http://en.wikipedia.org/wiki/Crofton_formula

which I thought was pretty neat.

[Dopefish]

Someone recently showed me a number (which I don't recall off the top of my head) which when multiplied by 2 or multiplied by 3 resulted in all the same digits, just rearranged.

Is there a name for numbers with that property? Is there anything fancy being done 'under the hood' that makes numbers like that behave in that manner, or is it just a kind of neat observation but without particular mathematical significance?

[jaap]

Someone recently showed me a number (which I don't recall off the top of my head) which when multiplied by 2 or multiplied by 3 resulted in all the same digits, just rearranged.

Is there a name for numbers with that property? Is there anything fancy being done 'under the hood' that makes numbers like that behave in that manner, or is it just a kind of neat observation but without particular mathematical significance?

It was probably a Cyclic number.

[ConMan]

Someone recently showed me a number (which I don't recall off the top of my head) which when multiplied by 2 or multiplied by 3 resulted in all the same digits, just rearranged.

Is there a name for numbers with that property? Is there anything fancy being done 'under the hood' that makes numbers like that behave in that manner, or is it just a kind of neat observation but without particular mathematical significance?

I think you're thinking of cyclic numbers. They're interesting, but most of the time numbers that only have a particular property in a single base are less useful than those where the property is independent of the base.

[Dopefish]

Yup, that's what I had in mind, and the first wiki example was what I was shown. Thanks, I'll give the wiki article a read through and see what insight it offers.

[jestingrabbit]

I thought this was pretty remarkable for the wide range of curves that come from the equation: e^{ait) - e^{bit}/2 + ie^{cit}/3.

http://tube.geogebra.org/student/m1292585

A favourite atm:

[LucasBrown]

Do I sense a fellow TMLT reader?

[jestingrabbit]

no

[roband]

Does anyone know of a tool which allows you to draw a curve onto a graph and will then calculate the equation of the curve you just drew?

I have no idea what I'm doing and I'm trying to calculate something which goes beyond my A-Level Maths education (which, I failed). Cheers.

[Eebster the Great]

I've seen tools that give curves of best fit of various types or exact polynomials for discrete data points, but nothing exactly like what you have described.

[Dason]

Can you describe what you're doing or possibly post the curve you're interested in?

[roband]

I believe I'm trying to create a equation in order to assign a score to something, based on numerous variables.

The curve I have shows the "acceptable" level for the score, based on 2 variables. The end outcome is that I want an equation to calculate the score based on those variables.

I have no idea if this makes any sense, but thanks for reading.


edit: here's the type of curve I want - http://imgur.com/5GHfeXd

only I want it to intersect the x-axis at x=4 and I want it to start running vertically at around x=1. Again, hopefully that makes sense.

[Eebster the Great]

y = cot(π/6(x-4))

[Flumble]

Other way around, Eebster. I'm guessing you did it from the top of your head.

y = cot(π/6(x-1))

[cyanyoshi]

How about this one? y = (x-4)/((x-1) (x-7))

[PM 2Ring]

Maybe something based on the inverse hyperbolic tangent?
Eg y = ln((7-x)/(x-1))/2