Say you're playing a game of Russian Roulette, but you can choose to aim the gun at your opponent once. Is there an optimal way to play?
[QUOTE=Krinkels;41606836]Say you're playing a game of Russian Roulette, but you can choose to aim the gun at your opponent once. Is there an optimal way to play?[/QUOTE]
How exactly are you playing?
If it's just 1/6 every time, the only optimal strategy is to avoid pointing it at your own head as long as possible! I suspect even if it's some more complex variation (loading more bullets as it goes on, or only spinning once at the start etc.) you're still better off not aiming at yourself until absolutely necessary...
So, since I'm not versed in discrete/computational math too much: is there an algorithm better than brute force to calculate, given a certain number of coins of different values, which combination to take to get a value as close as possible to a given amount? E.g. I have two quarters, three dimes, and a penny. What is the closest total value to 76 cents I can make out of some subset of those coins?
[QUOTE=JohnnyMo1;41896317]So, since I'm not versed in discrete/computational math too much: is there an algorithm better than brute force to calculate, given a certain number of coins of different values, which combination to take to get a value as close as possible to a given amount? E.g. I have two quarters, three dimes, and a penny. What is the closest total value to 76 cents I can make out of some subset of those coins?[/QUOTE]
I don't really have any experience with algorithms stuff, but it sounds like a variation on the subset-sum problem.
However if you just want to know the [i]closest[/i] value you can reach (rather than finding a way of getting a given sum) it might be different...
[QUOTE=Joey90;41946047]I don't really have any experience with algorithms stuff, but it sounds like a variation on the subset-sum problem.
However if you just want to know the [i]closest[/i] value you can reach (rather than finding a way of getting a given sum) it might be different...[/QUOTE]
I'd want the sum as well. Wikipedia has a link on their subset-sum page to the partition problem, which is very close to what I want. I was trying to figure out if there was an algorithm to always split a grocery list as closely as possible between two people. :v:
[editline]24th August 2013[/editline]
Apparently the problem is NP-hard. I guessed as much.
Does anybody know some good texts, lectures or problem sets for introductory linear algebra, and particularly (physical) Hilbert spaces? I've been following Shankar's Principles of Quantum Mechanics, and while I'm reasonably confident with the finite-dimensional cases, I get lost rapidly when he starts to use Hilbert spaces in either a classical or quantum context.
His mathematical introduction chapter is good, but I'm thinking seeing the material presented by someone else and having a few more exercises to figure it all out might help me wrap my head around it enough.
[QUOTE=JohnnyMo1;41947821]I'd want the sum as well. Wikipedia has a link on their subset-sum page to the partition problem, which is very close to what I want. I was trying to figure out if there was an algorithm to always split a grocery list as closely as possible between two people. :v:
[editline]24th August 2013[/editline]
Apparently the problem is NP-hard. I guessed as much.[/QUOTE]
An extreme long shot as I'm a mere physicist: so you have the Lagrange multipliers that emerge if you want to find the extrema of a function f with restrictions g (roughly, I don't know the specifics). Isn't there a discrete version of it that you can somehow throw into matlab?
I don't use matlab soooo
[editline]24th August 2013[/editline]
Numerical calculation? What am I, an engineer?
[editline]24th August 2013[/editline]
Also, I do not know if there is a discrete version of lagrange multipliers.
[QUOTE=JohnnyMo1;41951403]I don't use matlab soooo[/QUOTE]
Then out of curiosity, what do you use?
Or are you in the pen-and-paper realm of mathematics?
[QUOTE=nerdster409;41956626]Then out of curiosity, what do you use?
Or are you in the pen-and-paper realm of mathematics?[/QUOTE]
Pen and paper. Occasionally Mathematica to check long integrations. (Or do something silly like print that whole recently discovered Mersenne prime :v:)
[QUOTE=JohnnyMo1;41956705]Pen and paper. Occasionally Mathematica to check long integrations. (Or do something silly like print that whole recently discovered Mersenne prime :v:)[/QUOTE]
Yeah, since the University charges ten cents of my hard-earned money per printed page, I'm not gonna try that anytime soon.
[editline]25th August 2013[/editline]
I just took a look at the whole number.
Tell me: [B]Are you fucking crazy?![/B]
Print i.e. on the screen, not on paper.
If I did print it on paper, though, I would totally plaster it on the walls of the math building. I bet I could cover the whole thing.
How can I prove every convex function is continuous?
[QUOTE=Fourm Shark;42084105]I hate to revive a dead thread and basically ask someone to do my homework, but this one problem has been pissing me off.
Solve for x
4 = (x^0.5)/4 + x/1100
I have no idea how to solve this because no matter what I do, I end up with absurdly large numbers that I am not used to working with.
I tried clearing fractions first:
0 = (1100)(4)(x^0.5)/4 + (1100)(4)x/1100 - (1100)(4)4
0 = 1100(x^0.5) + 4x - 17600
I then tried to isolate the radical.
0 = [ 1100(x^0.5) + 4x - 17600 ]/1100
0 = x^0.5 + 4x/1100 - 16
-x^0.5 = 4x/1100 - 16
x^0.5 = -4x/1100 + 16
All I could think of doing now is squaring both sides to eliminate the radical x.
I end up with something pretty unmanageable.
x = (-4x/1100 + 16)^2
x = (16x^2)/1210000 -128x/1100 +256
Trying to work with it.
1100x = (17600x^2)/1210000 - 128x + 281600
0 = (17600x^2)/1210000 -1228x + 281600
Oh boy. Multiply by 1210000 to clear the fraction...
0 = 17600x^2 - 148588000x + 340736000000
Um, maybe divide by 800 to reduce a bit.
0 = 22x^2 - 185735x + 42592000
Ok, where do I go from here. Did I do something wrong? Is there another way to approach this problem?[/QUOTE]
That's the right method, you could save yourself some of the larger numbers by simplifying the fractions as they appear - for example when you had 17600/1210000, instead of multiplying everything by 1210000, simplify the fraction to 4/275 and multiply everything by 275 - much easier!
You also missed a couple of zeroes in your final simplification - you should have
0 = 22x^2 - 1857350x + 425920000
And because of that you can actually divide by 22 to get
0 = x^2 - 84425x + 19360000
which is a simple quadratic equation that you should be able to solve. Although you should note that since the original equation has a square root, assuming you are only taking positive square roots, only one of the solutions will work (so you should check which one does!)
Taking algebraic topology this semester. It's pretty awesome, except that the first homework has taken me ages.
My first university math course is tomorrow I'm so excited.
Also, JohnnyMo1, could you post homework like you used to? I couldn't understand much of them but they were still kinda neat to read.
[QUOTE=Krinkels;42093161]My first university math course is tomorrow I'm so excited.
Also, JohnnyMo1, could you post homework like you used to? I couldn't understand much of them but they were still kinda neat to read.[/QUOTE]
Longing for the golden days of this thread? Me too.
Sure. Finishing my algebraic topology homework now for tomorrow morning. Problem I'm stuck on is this:
[IMG]http://i43.tinypic.com/2rhoju8.png[/IMG]
Basically if you have some topological space that can be continuously shrunk to a point in itself, show that it can be continuously shrunk to ANY point in itself. I thought I had it this morning but I went to my professor's office hours and it turns out that one of my assumptions was wrong. I think I've got it now, sort of. Depends on me finishing the other problem I haven't finished, which will give me a proof that any map from an arbitrary space to a contractible space can be smoothly transformed into a constant map.
[editline]5th September 2013[/editline]
Surely if every map f:Y to X is nullhomotopic, then I can pick Y = X, and every constant map X to X must be homotopic, and homotopic to the identity map. Or am I crazy?
Hmm, nevermind. Don't think that's quite right...
Edit: Made a sacrifice to the math gods and they granted me a vision of a homotopy between the constant map to the point the space contracts to and an arbitrary constant map. And thus the problem was solved.
Just started my Discrete Math course. I got a question.
There's a problem about a mouse that intends to eat a 3x3x3 cube of cheese. It starts at a corner and eats 1x1x1, before going on to an adjacent one. So the question is, can the mouse end at the center?
The answer seems to be really simple if you know what an Eulerian walk is. However, is it correct to assume that this is a Eulerian walk problem?
A Eulerian walk is basically a walk where you traverse each road just once.
I think it's actually a Hamiltonian path problem. You have to visit each 1x1x1 cube exactly once, but you don't transverse all the possible connections between adjacent ones to do that.
[QUOTE=pebkac;42151961]I think it's actually a Hamiltonian path problem. You have to visit each 1x1x1 cube exactly once, but you don't transverse all the possible connections between adjacent ones to do that.[/QUOTE]
Ah of course! I'm so damn stupid. It makes so much sense lol
Now you can apply Ore's theorem to find out if it's Hamiltonian or not. His theorem states that for any graph G=(V,E) where the order is >= 3. Then if it fulfills:
[IMG]http://i.imgur.com/w8hJaWe.png[/IMG]
It is Hamiltonian.
The cube has 27 vertices. And it becomes quite obvious that no sum of the degree of two vertices not adjacent to each other in the cube will be equal or greater than 27.
You're not quite there yet. Ore's theorem is a sufficient, but not necessary condition for a graph to be Hamiltonian.
[editline]11th September 2013[/editline]
Actually, thinking a bit more about it, proving that a Hamiltonian cycle or path exists won't be enough due to the specific start- and endpoints.
Hmm yeah youre right Ill look more into it when i Get home. Just went on another lecture about graphs.
Curious...
When you go shopping and you have to work out a discount or $$/quantity..
Do you use a calculator (phone), or your noggin'?
[QUOTE=Bradyns;42162524]Curious...
When you go shopping and you have to work out a discount or $$/quantity..
Do you use a calculator (phone), or your noggin'?[/QUOTE]
In pretty much all our supermarkets the price/unit price/weight is written underneath the main price
[img]http://conversation.which.co.uk/wp-content/uploads/2013/05/sainsburys_apples_cropped.jpg[/img]
(or a 75p, 180g pack of something will say 41.7p per 100g etc.)
Since all our VAT is included in shelf prices we don't have to do that either. Pretty much the only thing I would have had to calculate would be 10% student discount, which I certainly can do in my head!
And I suspect even otherwise, I'd just 'approximately' work it out - if I had to work out some odd percentages for a purchase, I wouldn't whip out a calculator, I'd probably round to the nearest 5 or 10% and the price to 2 s.f. (unless I absolutely had to know)
Swebonny, a solution to your problem just randomly popped into my mind, and it's simpler than I thought.
Hint: [sp]Is the graph bipartite?[/sp]
What's purple and isomorphic to ZxZ?
[sp]The fundamental grape of the torus.[/sp]
Tonight's homework problem I was stuck on: show that if X is a space such that each point has a neighborhood which is star-convex, any path from x0 to x1 in the space is homotopic to a piecewise linear path from x0 to x1. Here's my solution:
[IMG]http://i40.tinypic.com/2m29h4x.png[/IMG]
90% sure there's a notational mistake in there somewhere and 60% sure there's reasoning mistake in there but I'm too tired and sick of writing it to reread it lol. Probably one of the most inelegant and handwavy proofs I've ever had to write. (but I went to my professor's office hours and got help so he knows my basic reasoning about it is solid)
Never thought I'd have to use the Lebesgue number lemma again.
[QUOTE=Fourm Shark;42173780]What level of math is that?[/QUOTE]
Algebraic topology. Grad level
What is the result of Problem 4?
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