## Search found 70 matches

Tue Jan 04, 2011 3:22 pm UTC
Forum: Mathematics
Topic: Sets of measure zero
Replies: 20
Views: 2615

### Re: Sets of measure zero

At this point, the only reasonably thing to think is "she lied". I don't think this explains the issue away. We could set up a similar situation where we pair up the numbers in the unit interval, and she tells you which pair was chosen. For instance, we could choose sets of the form {x,x+...
Tue Jan 04, 2011 2:52 pm UTC
Forum: Mathematics
Topic: Sets of measure zero
Replies: 20
Views: 2615

### Re: Sets of measure zero

@flangad - I never claimed that all sets of measure zero are countable? @jestingrabbit - I guess where my unease comes from is a situation like this. Suppose somebody pick a number randomly from the uniform distribution on [0,1]. She then tells me that it turned out to be a rational number. My gut i...
Tue Jan 04, 2011 1:28 pm UTC
Forum: Mathematics
Topic: Real numbers and uncountability.
Replies: 29
Views: 3175

### Re: Real numbers and uncountability.

I'm aware that some real numbers can't be expressed in decimal in a finite number of digits like 1/3 : is that what invalidates this? Yes. Your method only lists real numbers with finite decimal expansions, so numbers like 1/3 will never be listed. If you expand the method to list numbers that have...
Tue Jan 04, 2011 12:51 pm UTC
Forum: The Help Desk
Replies: 1
Views: 491

Hi all. I hope this is the right place to post this =) I recently sent an email to a university staff member using my university email address. Immediately afterwards, the staff member appeared in my Facebook friend suggestions. She and I have no Facebook friends in common. In fact, she only has sev...
Tue Jan 04, 2011 12:42 pm UTC
Forum: Mathematics
Topic: Sets of measure zero
Replies: 20
Views: 2615

### Sets of measure zero

Hi all. This is a thought that has been bugging me for some time, but I've never actually asked anyone about it. Consider the probability measure given by the the ordinary Lebesgue measure on the interval [0,1]. Any countable set has probability zero and so both the rational numbers in this interval...
Tue Apr 20, 2010 8:23 am UTC
Forum: Mathematics
Topic: Trig challenge
Replies: 20
Views: 2292

### Re: Trig challenge

As for the triple angle formula, I don't think it will help you out much, since there is no equivalent inverse formula (third angle formula). Simply using multiples of 3, 2, and 1/2, you will not get 1/180 from 1/5. This is true if you don't allow taking roots of complex numbers. The relevant secti...
Tue Apr 20, 2010 1:59 am UTC
Forum: Mathematics
Topic: Trig challenge
Replies: 20
Views: 2292

### Re: Trig challenge

Starting with \sin(\pi/5)=\frac14\sqrt{10-2\sqrt{5}} , you can use the double and triple angle formula and (hopefully not necessary) Cardano's method to find \sin(\pi/180) . This will certainly give you an answer. I'm not sure whether it will be in 'simplest form', partly because I d...
Tue Apr 20, 2010 1:39 am UTC
Forum: Science
Topic: Moon and sun in same part of sky
Replies: 17
Views: 2198

### Re: Moon and sun in same part of sky

Thanks guys. The interpretation 'left circle is the sun, bright patch is a reflection of the sun' sounds pretty plausible. Although the effect did seem fairly stable despite the clouds moving around.

I'm happy to have an alternative explanation for what I saw...
Mon Apr 19, 2010 2:06 pm UTC
Forum: Science
Topic: Moon and sun in same part of sky
Replies: 17
Views: 2198

### Moon and sun in same part of sky

I thought that somebody here might know the answer to this. I was in Estonia in January and while walking around I happened to see the sun and the moon visible in the same part of the sky. Photo: IMG_1016.jpg The photo was taken in the afternoon at about 14:30. Although it was overcast, the clouds w...
Mon Apr 19, 2010 1:54 pm UTC
Forum: Science
Topic: Could quantum computers generate random numbers?
Replies: 32
Views: 4492

### Re: Could quantum computers generate random numbers?

Mon Apr 19, 2010 1:50 pm UTC
Forum: Language/Linguistics
Topic: Sprechen Sie Deutsch bitte! [German practice]
Replies: 878
Views: 261311

### Re: Sprechen Sie Deutsch bitte! [German practice]

@ themonk (und Deutsche) Zum Thema deutsche Bücher, wie wär's mit Dürrenmatt? Ich fand 'Die Physiker' ganz witzig und nicht allzu schwierig. 'Der Richter und sein Henker' ist länger aber auch vom B2-C1 Niveau lesbar (oder so glaube ich - ich kenne die Niveaus erst seit ner Google-suche). 'Die Physik...
Mon Apr 19, 2010 12:47 pm UTC
Forum: Language/Linguistics
Topic: Supper or Dinner?
Replies: 55
Views: 8789

### Re: Supper or Dinner?

(Urban Australian) I would always say dinner for the evening meal, and never use this word for any other meal (no matter how large or grand). Having said that, I remember my grandparents sometimes saying supper for an evening meal. To me, this sounds very old-fashioned or 'English' (which are often ...
Mon Mar 29, 2010 9:19 am UTC
Forum: Mathematics
Topic: basic problem in graph theory
Replies: 3
Views: 817

### Re: basic problem in graph theory

You're right that the finiteness of the graph is important here. Think about the path v(1),v(2),...,v(k) that you constructed in part (a). What happens if v(k) is not adjacent to v(1)? As the degree of v(k) is at least k, you can extend the path by adding a new vertex v(k+1). How man...
Fri Mar 19, 2010 12:30 am UTC
Forum: Mathematics
Topic: Ideal Currency Denominations
Replies: 47
Views: 5776

### Re: Ideal Currency Denominations

I did a few computer runs of minimising the sum I put up earlier. The optimal denominations with n coins seem to hang around the powers of 100^(1/ n ), which I guess makes sense as with this distribution each denomination is doing "equal work". There are usually multiple optima though. Min...
Thu Mar 18, 2010 12:10 pm UTC
Forum: Mathematics
Topic: Ideal Currency Denominations
Replies: 47
Views: 5776

### Re: Ideal Currency Denominations

A related mathematical question is this: Suppose you can choose exactly n coin denominations, each less than a dollar. Let c(x) be the minimum number of coins needed to give x cents in change using these denominations. Which denominations should you choose to so that \sum_{x=1}^{99} c(x) is ...
Tue Nov 24, 2009 1:23 am UTC
Forum: News & Articles
Topic: Hadley Climatic RU hacked. Possibly damning data released.
Replies: 60
Views: 5173

### Re: Hadley Climatic RU hacked. Possibly damning data released.

I'm not sure if this interests anyone, but this issue was mentioned briefly in the Australian Senate yesterday: Senator Barnaby Joyce : Is the minister also aware of the director of the British Climatic Research Unit, Phil Jones, and emails that talk of ‘tricks’ to hide the decline in global warming...
Fri Feb 06, 2009 6:11 am UTC
Forum: Mathematics
Topic: Conditional probability / expectation
Replies: 2
Views: 790

### Re: Conditional probability / expectation

This is really helpful, thank you. I was a bit lost by it all. I might go and get myself Williams' book... Thanks again for your help!
Fri Jan 30, 2009 3:40 am UTC
Forum: Mathematics
Topic: Conditional probability / expectation
Replies: 2
Views: 790

### Conditional probability / expectation

I was hoping someone might help me understand the rigorous definitions of conditional probability and expectation. I hope I don't break the LaTeX support... Suppose we have a probability space (\Omega,\mathcal{F},\mathbb{P}) and a Borel random variable X:\Omega\to\mathbb{R} for which expecta...
Fri Jan 30, 2009 3:10 am UTC
Forum: Mathematics
Topic: Can someone spot my error
Replies: 2
Views: 543

### Re: Can someone spot my error

I haven't actually read all of your post, but the error comes in this step: n\ln(.33)\leq \ln(.01) \Rightarrow n\leq \frac{\ln(.01)}{\ln(.33)}. As \ln(.33) is negative, the inequality needs to be reversed. Edit: Incidentally, a much easier way to do this probl...
Sun Jan 25, 2009 11:57 pm UTC
Topic: The least racist country?
Replies: 107
Views: 16491

### Re: The least racist country?

Ok, I realise that there is no objective answer to "Which is the least racist country?". I was hoping that people might nominate what they think are sterling examples, with some evidence to back it up. Conclusively answering the question is not the point. Probably I should have anticipated...
Sun Jan 25, 2009 12:35 pm UTC
Topic: The least racist country?
Replies: 107
Views: 16491

### The least racist country?

In the light of discussions of racism in the US and in my own country (Australia), I've been thinking... What do people think is the least racist country? To qualify for consideration, a country must have a significant ethnic or cultural minority, or else that country's views can be considered untes...
Sun Jan 25, 2009 10:15 am UTC
Forum: News & Articles
Topic: My President is Black, Fox News Complains
Replies: 310
Views: 20417

### Re: My President is Black, Fox News Complains

Is it just me, but are some of the privileges from the white priviledge page linked to earlier only applicable if the person in question is themself racist? Not at all. I admit I find number 1 a little odd, but the others are closely related to societal (and self-) expectations. If almost all the p...
Sat Jan 24, 2009 5:50 am UTC
Forum: Mathematics
Topic: A dice game problem
Replies: 14
Views: 1140

### Re: A dice game problem

Not if you are talking about a unilateral change of strategy. If four of the players play the pure strategy k=78, then for the remaining player k=78 strictly dominates all other pure strategies. As mixed strategies are just linear combinations of pure strategies, this means that no other strategy, p...
Sat Jan 24, 2009 1:57 am UTC
Forum: Mathematics
Topic: A dice game problem
Replies: 14
Views: 1140

### Re: A dice game problem

Yup, Cosmologicon is right. A big nasty excel file shows that there is a pure strategy Nash equilibrium when each player rolls again if their roll is less than 78. (I have assumed that in the case of a tie, the game just restarts.) Cosmo's proof should be enough for everyone though, right? I don't w...
Sat Jan 24, 2009 12:13 am UTC
Forum: Mathematics
Topic: Random walks on the roots of unity
Replies: 10
Views: 2470

### Re: Random walks on the roots of unity

Thanks, t0rajir0u and antonfire, that was really very helpful. I used your suggestions to write out a proof that looks essentially like antonfire's post above (which I've only just seen). It took me a while to work out that there is an integral basis for \mathbb{Z}[\omega_n] , though =) @antonfire: ...
Fri Jan 23, 2009 12:06 pm UTC
Forum: Mathematics
Topic: Oh Linear Algebra, how I curse thee
Replies: 54
Views: 5563

### Re: Oh Linear Algebra, how I curse thee

An isomorphism isn't necessarily linear. This is only true on a vector space. As many others have said, an isomorphism is a bijective function that preserves some structure of the underlying space. Exactly what structure depends on the space in question. So, for a vector space, the important structu...
Fri Jan 23, 2009 4:30 am UTC
Forum: Language/Linguistics
Topic: Periods in quotations?
Replies: 5
Views: 1508

### Re: Periods in quotations?

Bwahaha Esquilax that is the funniest avatar i've seen in a long time! =) Carry on...
Fri Jan 23, 2009 4:02 am UTC
Forum: Mathematics
Topic: Random walks on the roots of unity
Replies: 10
Views: 2470

### Re: Random walks on the roots of unity

the relevant linear dependence involving \phi(n) roots of unity is given by \Phi_n(\omega)=0. I don't geddit... Doesn't \Phi_n(\omega)=0 only hold when \omega is a primitive n th root? Or does this notation mean something else? It's non-trivial to prove, but a return to the ...
Fri Jan 23, 2009 2:54 am UTC
Forum: Mathematics
Topic: Fermat's Last Theorem...
Replies: 42
Views: 3719

### Re: Fermat's Last Theorem...

In order to acknowledge the validity of a proof, in principle, you should understand exactly how it works well enough to be certain of its correctness. In practice... In some ways, I think that this reflects a change in mathematical culture that began in (about?) the second half of the eighteenth c...
Fri Jan 23, 2009 1:51 am UTC
Forum: Mathematics
Topic: Random walks on the roots of unity
Replies: 10
Views: 2470

### Re: Random walks on the roots of unity

antonfire wrote:The sums of the n'th roots of unity for n>1 form a lattice of dimension phi(n)

If you don't mind me asking, how do you define the dimension of a lattice? There's a few different definitions and I'm not sure which one would make sense here.
Thu Jan 22, 2009 11:55 am UTC
Forum: Mathematics
Topic: Argh... what am I not seeing? (homework problem)
Replies: 14
Views: 2044

### Re: Argh... what am I not seeing? (homework problem)

If y=e^x then e^(-x)=1/y.
Thu Jan 22, 2009 6:43 am UTC
Forum: Mathematics
Topic: Random walks on the roots of unity
Replies: 10
Views: 2470

### Re: Random walks on the roots of unity

The Pathological Case wrote:Do you mean the particle moves towards one of the roots of unity, which are fixed around the unit circle, or the particle's position is displaced by one of the roots of unity each second?

The latter. I'll edit my post to make this clearer.
Thu Jan 22, 2009 4:28 am UTC
Forum: Mathematics
Topic: Random walks on the roots of unity
Replies: 10
Views: 2470

### Random walks on the roots of unity

Ok, here is a puzzle question. I know the answer already, but I am not very satisfied with the unity of my proof. Maybe someone here has a better approach? Imagine a particle starting at the origin in the complex plane. Each second, the particle's position is displaced by one of the n th roots of un...
Fri Jan 16, 2009 4:35 am UTC
Topic: Moral Facts: their existence and nature
Replies: 22
Views: 5595

### Re: Moral Facts: their existence and nature

I don't think the OP's classification scheme is too bad. It's not comprehensive, but it might make people think harder about what they are saying. In the cognitivist / non-cognitivist debate, I think there are two questions that are easily (and often) confused. The first is What do we mean when we m...
Sat Jan 10, 2009 1:08 pm UTC
Forum: Mathematics
Topic: Prove that two curves in a square intersect
Replies: 76
Views: 6674

### Re: Prove that two curves in a square intersect

Ended wrote:Because {x} and {y} (the images of the curves) are compact, E>0 iff the curves don't intersect.

Ok, thank you Ended. Looks like all I did was reprove that any two compact sets have a nearest point =) Can't hurt, I guess...
Sat Jan 10, 2009 11:00 am UTC
Forum: Mathematics
Topic: Prove that two curves in a square intersect
Replies: 76
Views: 6674

### Re: Prove that two curves in a square intersect

Ok, I'm pretty sure this proof doesn't work, but I'd like someone to explain to me why. Let E=\inf|x-y| , where x and y are vector-valued points that range respectively over the two curves. If E>0, then I assume we can derive a contradiction by taking an open set around the curve (I haven't proved t...
Mon Dec 29, 2008 3:07 pm UTC
Forum: Mathematics
Topic: Milkshakes
Replies: 5
Views: 1348

### Re: Milkshakes

I think this is a pretty hard question, but I'll make a start. Consider a single cylindrical glass/can. The glass has radius R and the volume of liquid in it is given by some function V(t) . The open top of the glass has coefficient of cooling \rho_1 , the bottom has \rho_2 and the sides hav...
Wed Dec 24, 2008 10:13 pm UTC
Forum: Mathematics
Topic: Traffic lights (optimal braking)
Replies: 12
Views: 2062

### Re: Traffic lights (optimal braking)

Yay =) This is the same answer I came to analytically. Suppose v(t) is the velocity, d is the distance, and the time until the light changes is uniform on (0,c).Then let T be the earliest time such that \int_0^T v(t)\ dt= d. For the braking to be efficient, we must have T\leq c . But the exp...
Wed Dec 24, 2008 12:39 pm UTC
Forum: Mathematics
Topic: Fun probability things?
Replies: 42
Views: 3446

### Re: Fun probability things?

Stuff about cars and keys... What about this variant? There are seven cars and seven matching keys. Each player picks a key out of the hat. The first player chooses a car and tries to start it. If they are successful, they win the car and the game ends. If not, the next player gets to try their key...
Tue Dec 23, 2008 2:59 am UTC
Forum: Mathematics
Topic: Traffic lights (optimal braking)
Replies: 12
Views: 2062

### Traffic lights (optimal braking)

Ok, I've been thinking about this problem for years, but I've never given it a serious go. Suppose it's late at night and you're driving up to a traffic light. The light is red, but you know it's on a fixed two-state cycle that alternates every thirty seconds. You don't know when the light last chan...