Search found 44 matches

by DT_
Mon Feb 07, 2011 6:44 am UTC
Forum: Computer Science
Topic: How do I show a DFA doesn't exist?
Replies: 4
Views: 2074

Re: How do I show a DFA doesn't exist?

Ah ... so I could say: To accept or reject a string, the DFA must remember what the nth-to-last character is. When a character is read, the DFA should transition to a final state iff what was previously the n-1st character from the end is a 1. When a second character is read, the DFA should transiti...
by DT_
Mon Feb 07, 2011 5:57 am UTC
Forum: Computer Science
Topic: How do I show a DFA doesn't exist?
Replies: 4
Views: 2074

How do I show a DFA doesn't exist?

Usually when I want to show an object doesn't exist, I would assume that it does and try to arrive at a contradiction. However, I can't really make that approach work for showing that a particular DFA doesn't exist. Any suggestions on how I might go about formally proving the following (homework) pr...
by DT_
Tue Nov 16, 2010 8:09 pm UTC
Forum: Mathematics
Topic: Question on the sum of exponential distributions
Replies: 2
Views: 734

Re: Question on the sum of exponential distributions

Ah, it should have been [imath]\Pr\left(X + Y \le k\right) = \int_0^k \! e^{-x}\left(1 - e^{-\left(k-x\right)}\right) \, \mathrm{d}x[/imath]. Thank you very much!
by DT_
Tue Nov 16, 2010 6:55 pm UTC
Forum: Mathematics
Topic: Question on the sum of exponential distributions
Replies: 2
Views: 734

Question on the sum of exponential distributions

I have the following homework problem: Let X and Y be independent, exponentially distributed random variables with parameter 1. Find the density function and distribution function for X + Y. I'd like to solve it using: \Pr\left(X + Y \le k\right) = \int_0^\infty \! f(x)\Pr\left(Y...
by DT_
Mon Nov 08, 2010 8:37 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0816: "Applied Math"
Replies: 88
Views: 28554

Re: 0816: Applied Math

To be a small sticker on notation: where it says P \Lambda \bar{P} , what you probably meant, the adjoint (conjugate transpose for finite-dimensional spaces), is usually written P^* rather than \bar{P} to distinguish it from only doing complex conjugate (without doing transpose) of all the elements...
by DT_
Tue Apr 27, 2010 2:43 pm UTC
Forum: Mathematics
Topic: Can someone explain this problem statement?
Replies: 6
Views: 910

Re: Can someone explain this problem statement?

Couldn't quite show all those things but thank you for your help :)
by DT_
Tue Apr 27, 2010 1:05 pm UTC
Forum: Mathematics
Topic: Can someone explain this problem statement?
Replies: 6
Views: 910

Re: Can someone explain this problem statement?

Hmm, I've tried everything I can think of but I don't see how I can show how closely together I have to pick my points if I don't even know what my function is.
by DT_
Tue Apr 27, 2010 12:07 pm UTC
Forum: Mathematics
Topic: Can someone explain this problem statement?
Replies: 6
Views: 910

Re: Can someone explain this problem statement?

Yeah, that's the metric we're using. So I want to first figure out what the partial sums converge to, and then once I know that function f, show it's uniformly continuous by showing that for any epsilon > 0, there's a global delta > 0 such that for all x and y with |x - y| < delta, |f(x) - f(y)| < e...
by DT_
Tue Apr 27, 2010 11:57 am UTC
Forum: Mathematics
Topic: Can someone explain this problem statement?
Replies: 6
Views: 910

Can someone explain this problem statement?

I can't seem to understand what it is that I have to prove for this problem, let alone how to prove it. The problem is as follows: For each n \ge 0 , consider the function f_n:R \rightarrow R defined by the formula f_n = 3^{-n}\sin(2^n x) . Show that \sum^\infty_{n=0} f_n converges to a unif...
by DT_
Tue Apr 13, 2010 2:23 pm UTC
Forum: Mathematics
Topic: Need help on analysis problem
Replies: 21
Views: 1947

Re: Need help on analysis problem

Awesome, thanks so much for the help. Finished with 25m to spare :)
by DT_
Tue Apr 13, 2010 1:56 pm UTC
Forum: Mathematics
Topic: Need help on analysis problem
Replies: 21
Views: 1947

Re: Need help on analysis problem

Ok, that makes sense. Then would the following be a valid way of showing the function is well defined? Since {x_n} and {y_n} both converge to p, there exists an N for which all n >= N implies d({x_n}, p) < delta, d({y_n}, p) < delta. Now since f is uniformly continuous, this implies that d(f(x_n), f...
by DT_
Tue Apr 13, 2010 1:37 pm UTC
Forum: Mathematics
Topic: Need help on analysis problem
Replies: 21
Views: 1947

Re: Need help on analysis problem

Ended wrote:This will then automatically imply continuity on the boundary.

Could you explain this a bit further please?
by DT_
Tue Apr 13, 2010 1:32 pm UTC
Forum: Mathematics
Topic: Need help on analysis problem
Replies: 21
Views: 1947

Re: Need help on analysis problem

Harg wrote:Metric spaces are first-countable, so f is continuous iff it preserves limits of sequences. This should be enough for your problem, I think.

That may be true but we haven't defined first-countable spaces so I can't use that in a proof.
by DT_
Tue Apr 13, 2010 12:11 pm UTC
Forum: Mathematics
Topic: Need help on analysis problem
Replies: 21
Views: 1947

Need help on analysis problem

Yes, this is homework. I've been bashing my head against this for a long time, and it's due in 3 hours. Let X and Y be metric spaces and E a subset of X. Show that if Y is complete and f is a uniformly continuous function from E to Y then f can be extended to a continuous function from the closure o...
by DT_
Fri Mar 12, 2010 4:42 am UTC
Forum: Mathematics
Topic: Rational^irrational can yield irrational
Replies: 27
Views: 8728

Re: Rational^irrational can yield irrational

supremum wrote:This step is wrong. It's equal to [imath]4^{\sqrt{2}}[/imath]

Oops, that's embarrassing heh >.<
by DT_
Fri Mar 12, 2010 3:41 am UTC
Forum: Mathematics
Topic: relation which is symmetric, reflexive but not transitive?
Replies: 20
Views: 24185

Re: relation which is symmetric, reflexive but not transitiv

That isn't what I'm trying to prove though. I've been told that if X (a set) has only two elements, there are three combinations of properties that cannot happen for a relation R. One combination is reflexive and symmetric but not transitive. I'm then asked to find the other two. The given combinat...
by DT_
Fri Mar 12, 2010 3:30 am UTC
Forum: Mathematics
Topic: relation which is symmetric, reflexive but not transitive?
Replies: 20
Views: 24185

Re: relation which is symmetric, reflexive but not transitiv

so by definition all relations that are reflexive are also symmetric? No, a relation can be reflexive and not symmetric. For instance, if you have elements {a, b} and the relation is {(a, a), (a, b), (b, b)}, then the relation is reflexive and not symmetric. However, for the particular case of {(a,...
by DT_
Fri Mar 12, 2010 3:16 am UTC
Forum: Mathematics
Topic: relation which is symmetric, reflexive but not transitive?
Replies: 20
Views: 24185

Re: relation which is symmetric, reflexive but not transitiv

if you have a set A = { a, b} then a relation which is reflexive and symmetric is: {(a,a),(b,b),(a,b),(b,a)} but surely by definition that is transitive as well, since for a relation to be transitive, a = b = c, and in that relation, a maps to b which maps to a which maps to a i.e. a -> b b -> a a ...
by DT_
Fri Mar 12, 2010 2:58 am UTC
Forum: Mathematics
Topic: Rational^irrational can yield irrational
Replies: 27
Views: 8728

Re: Rational^irrational can yield irrational

He has to provide an example and prove that the example is indeed irrational, so a nonconstructive proof wouldn't do much good.

edit: erroneous proof.
by DT_
Fri Mar 05, 2010 11:07 pm UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

I don't think doing (3x+1)/2 infinitely to any natural number will continually result in a natural number. Take any odd number that works for the Collatz conjecture, for example, 11: doing (3x+1)/2 infinitely gives you 11-->17-->26-->14.5, which is not a natural number and will only get worse from ...
by DT_
Fri Mar 05, 2010 8:20 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

Noooooooo!. Now I have only two brain cells left, and I was hording that one for some future debauchery! :) <------ is no kind of mathematician , but , thanks for the thought anyway. Imagine you have a sequence of dominoes. Suppose you could show that if the k-th domino were to fall, then the k + 1...
by DT_
Fri Mar 05, 2010 8:06 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

So, as far as I can tell, we're basically trying to find whether there exists an odd number that can be put through (3x+1)/2, or 1.5x+.5, infinitely without becoming a non-natural number. Given any natural number as input, the sequence of numbers produced by the Collatz conjecture for that natural ...
by DT_
Fri Mar 05, 2010 7:42 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

Thank you for that, it was very clearly expressed. And covered what I saw ( using my three brain cells till they squealed ) as the problem, you could never prove it correct , because of an infinity of potential numbers, right? Not quite - it is possible to prove results for infinitely many numbers ...
by DT_
Fri Mar 05, 2010 7:19 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

If you multiply any odd number by 3 and add 1 it become even. Correct. If you multiply any even number by 3 and add 1 it become odd. True in general but if n is even then the next value in the sequence will be n/2, not (3n + 1). If you divide any even number by 2, and keep dividing by 2 you will ev...
by DT_
Fri Mar 05, 2010 6:39 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

chocolate.razorblades wrote:It seems like graphs would make suitable proofs for these, but there are already a lot and I don't understand why the math community won't accept them.

No one has yet proven that given the graph as constructed in the comic and any natural number n, that n is in the graph.
by DT_
Fri Mar 05, 2010 6:31 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

but if you do "n*3+1", then you can get odds from odds An odd number times an odd number + 1 is always even, so you can't get odds from odds. However, since (3n+1)/2 is larger than n, it isn't trivial to prove that n will always reach 1. For the example above where you simply do n + 1 for...
by DT_
Fri Mar 05, 2010 6:24 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

eviloatmeal wrote:Ok, so is there a simple explanation why odds need to be multiplied by 3? It seems to work by just adding one, and logically, it should, no?

The function was not made with the goal of reaching 1. It was simply discovered by Collatz who conjectured that it always reached 1.
by DT_
Fri Mar 05, 2010 6:19 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0710: "Collatz Conjecture"
Replies: 278
Views: 71884

Re: "Collatz Conjecture" Discussion

Shouldn't there be an arrow from 1 to 4?

No, once you reach 1 you're done.
by DT_
Sun Feb 21, 2010 12:58 am UTC
Forum: Mathematics
Topic: Creating a function (Work problem!)
Replies: 12
Views: 1674

Re: Creating a function (Work problem!)

LaTex was indeed the problem, and I must admit, after getting LaTeX working, it's damned nice to see actual math formulas on the computer. By the way; brilliant. I would have given up far before trying to implement a high order polynomial. I don't know why, but my mindset was suck on using trigonom...
by DT_
Fri Feb 19, 2010 11:15 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0704: "Principle of Explosion"
Replies: 104
Views: 32520

Re: "Principle of Explosion" Discussion

In other words, if someone tells me "A implies B", and I say "fair enough, but look - A is false". They say "but if A was true, B would be true too!", to which I reply "maybe so, but perhaps if A was true B would be false - since A is false, it's impossible to tel...
by DT_
Fri Feb 19, 2010 10:53 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0704: "Principle of Explosion"
Replies: 104
Views: 32520

Re: "Principle of Explosion" Discussion

I'm still lost. I don't see how A being false makes the statement "If A then B" automatically true. Doesn't it just mean that we don't know if the statement is true or not? Sort of like this: - "If A then B" or "A implies B" means that B must be true if A is true, but ...
by DT_
Fri Feb 19, 2010 6:15 am UTC
Forum: Mathematics
Topic: Creating a function (Work problem!)
Replies: 12
Views: 1674

Re: Creating a function (Work problem!)

DT_, I might just be being dull, but I'm not sure what you're suggesting. The formatting is a little beyond me. Cheers! I'm suggesting that you use the function g(x) = \left(−1.6703133632006x^3 + 2.6946968137368x^2 − 0.89531845609866x + 1.1195795491556\right) / A , from \pi/8 to \pi...
by DT_
Thu Feb 18, 2010 10:44 pm UTC
Forum: Mathematics
Topic: Creating a function (Work problem!)
Replies: 12
Views: 1674

Re: Creating a function (Work problem!)

Ok, actually got it this time: f(x) = 1 / \left(A*\cos\left(x\right)\right) Choose g(x) = C_3x^3 + C_2x^2 + C_1x + C_0 . You have the following 4 constraints: 1. g(\pi/8) = f(\pi/8) 2. g'(\pi/8) = f'(\pi/8) 3. g''(\p...
by DT_
Thu Feb 18, 2010 1:14 pm UTC
Forum: Mathematics
Topic: Creating a function (Work problem!)
Replies: 12
Views: 1674

Re: Creating a function (Work problem!)

Isn't that comparing apples and oranges? Saying that the derivative and second derivative are equal, I mean -- they are in different units ? Let f(x) = A\sec(x) and g(x) = your function. He's saying that f'(\pi/8) = g'(\pi/8) and f''(\pi/8...
by DT_
Thu Feb 18, 2010 10:05 am UTC
Forum: Mathematics
Topic: Creating a function (Work problem!)
Replies: 12
Views: 1674

Re: Creating a function (Work problem!)

edit: obsolete post. See my post below.
by DT_
Fri Feb 12, 2010 3:34 am UTC
Forum: Mathematics
Topic: Why is this true?
Replies: 12
Views: 1772

Re: Why is this true?

That clears it up, thanks Qaanol.
by DT_
Fri Feb 12, 2010 2:18 am UTC
Forum: Mathematics
Topic: Why is this true?
Replies: 12
Views: 1772

Re: Why is this true?

@hnooch: Do you mean f(z) = ([imath]z^{n/2}[/imath] - 1)([imath]z^{n/2}[/imath] + 1) (if n is even), and then continue reducing those until you somehow get some result?

@MartianInvader: Unless I'm way off about your alternate approach, I think you misunderstood my question.
by DT_
Fri Feb 12, 2010 1:30 am UTC
Forum: Mathematics
Topic: Why is this true?
Replies: 12
Views: 1772

Re: Why is this true?

@Kurushimi: Yes, it should be n(n-1)/2. @antonfire/Qaanol: Up until now I hadn't even heard of a primitive root of unity. It isn't in the textbook and the professor never talked it, we were just given that \omega=e^{2\pi i / n} (I should mention that this isn't a math course, it's an algorithms cour...
by DT_
Thu Feb 11, 2010 11:50 pm UTC
Forum: Mathematics
Topic: Why is this true?
Replies: 12
Views: 1772

Why is this true?

The problem asks me to find the product of the n nth-roots of unity, when n is odd and n is even. I have \omega^0\times\omega^1\times\dots\times\omega^{n-1} = \omega^{\frac{n(n + 1)}{2}} . If n is odd then I've shown the answer is 1. If n is even then I know that the answer is -1 but I can't...
by DT_
Mon Feb 01, 2010 10:30 am UTC
Forum: Mathematics
Topic: HW question regarding supremums
Replies: 13
Views: 1060

Re: HW question regarding supremums

Thanks for pointing out the infinity case.

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