Search found 825 matches

by Torn Apart By Dingos
Sun Nov 09, 2008 9:42 am UTC
Forum: Mathematics
Topic: Help an Aussie engineer pass his Numerical Analysis Exam!
Replies: 3
Views: 909

Re: Help an Aussie engineer pass his Numerical Analysis Exam!

It seems like all you need to do is to express a higher order differential equation as a system of linear differential equations. For example, y''+2y'+3y=0 is equivalent to the system u'+2u+3y=0 y'=u. The principle is the same for higher order ODEs: just introduce some new functions u=y', v=y''=u', ...
by Torn Apart By Dingos
Sun Nov 09, 2008 6:26 am UTC
Forum: Logic Puzzles
Topic: Tetris Puzzle
Replies: 13
Views: 3805

Re: Tetris Puzzle

I read that Wikipedia page a while back, and it linked to the paper How to Lose at Tetris , which no longer seems to be referenced. It presents a nifty proof that you will inevitably lose if you are given an alternating sequence of S and Z tetrominoes (the odd numbered pieces are Ss, the even number...
by Torn Apart By Dingos
Fri Nov 07, 2008 9:49 pm UTC
Forum: Mathematics
Topic: Help with an infinite series.
Replies: 9
Views: 1469

Re: Help with an infinite series.

What that example illustrates is that you can't rearrange the terms in a series and expect the result to be the same. Your series can actually be rearranged so as to converge to any given value (say 17) or even diverge to infinity.
by Torn Apart By Dingos
Thu Nov 06, 2008 10:51 pm UTC
Forum: Mathematics
Topic: Frustrating "Permutations-Type" Balls-and-Containers Problem
Replies: 7
Views: 1138

Re: Frustrating "Permutations-Type" Balls-and-Containers Problem

tl;dr.

Did you try the online encyclopedia of integer sequences? That's always a good place to start. There are formulas if you scroll down.
http://www.research.att.com/~njas/seque ... &go=Search
by Torn Apart By Dingos
Tue Nov 04, 2008 11:08 pm UTC
Forum: Mathematics
Topic: Are "sometimes" statements true or false?
Replies: 16
Views: 2033

Re: Are "sometimes" statements true or false?

You win this round!
by Torn Apart By Dingos
Tue Nov 04, 2008 9:52 pm UTC
Forum: Mathematics
Topic: Are "sometimes" statements true or false?
Replies: 16
Views: 2033

Re: Are "sometimes" statements true or false?

I don't read the "a" in that sentence as a quantifier at all. Compare to "a continuous function is differentiable", which I don't think anyone would interpret as "(at least) one continuous function is differentiable". This is more of a language problem than a logical on...
by Torn Apart By Dingos
Mon Nov 03, 2008 3:26 pm UTC
Forum: Mathematics
Topic: Looking for interesting math books at the high school level
Replies: 3
Views: 940

Re: Looking for interesting math books at the high school level

There's not really any reason you couldn't read an introductory university level book. Books on discrete math and algebra in particular start entirely from scratch and don't assume any preexisting knowledge. "Discrete Mathematics" by Biggs is a gentle and readable book that is fun to read.
by Torn Apart By Dingos
Sun Nov 02, 2008 11:59 pm UTC
Forum: Mathematics
Topic: How to state continuity for some odd functions
Replies: 12
Views: 1645

Re: How to state continuity for some odd functions

Yakk wrote:A function f: A->B is called continuous at a point x if there is a neighborhood N of x such that f restricted to N is continuous.

Counterexample: f(x)=x if x is rational, f(x)=0 if x is irrational, continuous at zero but in no neighbourhood of zero.
by Torn Apart By Dingos
Sun Nov 02, 2008 5:27 pm UTC
Forum: Mathematics
Topic: For those who like lesbians and fractals
Replies: 5
Views: 2018

Re: For those who like lesbians and fractals

rho wrote:Lesbians at 3500 K? ...If this were /b/ there'd be pictures.

Image
(The sun is female, right?)
by Torn Apart By Dingos
Sun Nov 02, 2008 10:13 am UTC
Forum: Mathematics
Topic: For those who like lesbians and fractals
Replies: 5
Views: 2018

Re: For those who like lesbians and fractals

The hotness radiating from n lesbian bodies (which to great accuracy may be approximated as black bodies) was first given by Planck in the form of Planck's law: u(\lambda,T) = {8\pi h c\over \lambda^5}{1\over e^{\frac{h c}{\lambda kT}}-1}. You're right in that the hotness eventually decays, ...
by Torn Apart By Dingos
Sat Nov 01, 2008 3:32 pm UTC
Forum: Mathematics
Topic: "What am I missing?"|"Phantom Root."
Replies: 2
Views: 864

Re: "What am I missing?"|"Phantom Root."

When you equated the two sides, you took the square of both of them to get rid of the square root, right? That introduces new roots. For instance, x=1 is not equivalent to x^2=1. The latter has two solutions: x=1 and x=-1. What happens when you square both sides of an equation is that they become eq...
by Torn Apart By Dingos
Tue Oct 28, 2008 8:48 pm UTC
Forum: Mathematics
Topic: Proof of Taylor expansion of trigonometric functions
Replies: 6
Views: 2095

Re: Proof of Taylor expansion of trigonometric functions

Historically (afaik), e^(ix)=cos(x)+isin(x) was proved from those series (well, actually defined from it, but the series expansion suggested this to be the best definition). To get the series, just differentiate and plug into the Taylor series. The values of cos(0), cos'(0), cos''(0), cos'''(0), ......
by Torn Apart By Dingos
Mon Oct 27, 2008 12:04 am UTC
Forum: Mathematics
Topic: Matrices and roots of equations
Replies: 13
Views: 1647

Re: Matrices and roots of equations

I interpreted his "polymatrix" as a polynomial in which both the variable and the coefficients are matrices.
by Torn Apart By Dingos
Sun Oct 26, 2008 12:49 pm UTC
Forum: Mathematics
Topic: Matrices and roots of equations
Replies: 13
Views: 1647

Re: Matrices and roots of equations

Does this mean that any "polymatrix" (yes I just made that word up) equation has an infinite amount of matrix roots? Or just this one? No, X^2= \begin{pmatrix}1&0\\0&-1\end{pmatrix} has no solutions for real 2x2-matrices, and only finitely many for complex 2x2-matrices: if X has e...
by Torn Apart By Dingos
Sat Oct 18, 2008 8:36 pm UTC
Forum: Mathematics
Topic: Balls in balls in R^n
Replies: 25
Views: 4139

Re: Balls in balls in R^n

Thanks for the update! That problem seems more reasonable.
by Torn Apart By Dingos
Wed Oct 08, 2008 10:08 am UTC
Forum: Mathematics
Topic: Acceleration troubles
Replies: 9
Views: 1353

Re: Acceleration troubles

Oh snap!
by Torn Apart By Dingos
Wed Oct 08, 2008 3:23 am UTC
Forum: Mathematics
Topic: Homework help needed (problem from Spivak's Calculus)
Replies: 26
Views: 4402

Re: Homework help needed (problem from Spivak's Calculus)

Right, the terms cancel. I agree that's slicker.
by Torn Apart By Dingos
Tue Oct 07, 2008 5:53 pm UTC
Forum: Mathematics
Topic: Homework help needed (problem from Spivak's Calculus)
Replies: 26
Views: 4402

Re: Homework help needed (problem from Spivak's Calculus)

What I did with Yakk's hint was: WLOG assume x<0<y (if they have the same sign, it's trivial), and let k=x/y. Divide both sides of the inequality by y^4, and we get k^4+k^3+k^2+k+1>0, or equivalently (1-k^5)/(1-k)>0, which certainly is true because k<0. Edit: Sorry, I don...
by Torn Apart By Dingos
Fri Oct 03, 2008 10:01 pm UTC
Forum: Mathematics
Topic: Arc of an ellipse
Replies: 10
Views: 1759

Re: Arc of an ellipse

There's no explicit formula for that either. If you want to approximate it, start at the point and walk tiny straight line steps along the curve until you've walked the given arc length.
by Torn Apart By Dingos
Fri Oct 03, 2008 8:38 pm UTC
Forum: Mathematics
Topic: Arc of an ellipse
Replies: 10
Views: 1759

Re: Arc of an ellipse

There isn't a nice, neat formula for the arc length. Wikipedia gives the formula for it in terms of the elliptic integral, which basically is saying it's equal to itself. Some approximations are listed though. If you want to find it with code, you can simply connect some points on the arc together w...
by Torn Apart By Dingos
Thu Sep 25, 2008 7:14 pm UTC
Forum: Mathematics
Topic: Naughty Functions
Replies: 59
Views: 6148

Re: Naughty Functions

The indicator function function of the rationals. Nowhere continuous. The popcorn function, continuous only on the irrationals. The Busy Beaver function. Bijections from N to N^2. One that is given by a formula is f(n,m)=(n+m)*(n+m+1)/2+n (it's enumerates the elements on the diagonals). I haven't se...
by Torn Apart By Dingos
Wed Sep 24, 2008 11:32 am UTC
Forum: Mathematics
Topic: Question about Infinity
Replies: 72
Views: 5271

Re: Question about Infinity

Your argument there doesn't work at all. Since there is also no *rational* number that is closest to 1. And yet there's the same cardinality of rational numbers as there is of integers or whole numbers. Being dense is different from being complete. I disagree.....because between any two rationals t...
by Torn Apart By Dingos
Wed Sep 17, 2008 4:19 pm UTC
Forum: Mathematics
Topic: Down with the Sickness.
Replies: 12
Views: 1822

Re: Down with the Sickness.

Someone else thought of that same rain question and I answered it here: http://forums.xkcd.com/viewtopic.php?f=7&t=318&p=4533 You may be onto something about math being a disease. The math building in my local university reads "Pathologic Bacteriology" in big letters above the entr...
by Torn Apart By Dingos
Wed Sep 17, 2008 8:39 am UTC
Forum: Mathematics
Topic: Mathematics and Beauty
Replies: 42
Views: 5523

Re: Mathematics and Beauty

antonfire wrote:Eh, that's pretty much the same proof, only in base 3.
If you just say "let I_{n+1} a subinterval of I_n that x_n is not in" instead of explicitly dividing each interval into three parts, which is really only done to avoid implicitly using the axiom of choice, it's not very similar anymore.
by Torn Apart By Dingos
Tue Sep 16, 2008 5:44 pm UTC
Forum: Mathematics
Topic: Randomly Generated Images
Replies: 26
Views: 4394

Re: Randomly Generated Images

I don't know, things didn't go so well for that guy in Hollow Man.
by Torn Apart By Dingos
Tue Sep 16, 2008 5:33 pm UTC
Forum: Mathematics
Topic: Randomly Generated Images
Replies: 26
Views: 4394

Re: Randomly Generated Images

Shall I use my invisibility to fight crime or for evil?
by Torn Apart By Dingos
Tue Sep 16, 2008 5:30 pm UTC
Forum: Mathematics
Topic: Mathematics and Beauty
Replies: 42
Views: 5523

Re: Mathematics and Beauty

I'm a fan of the proof that the reals are uncountable. There are several such proofs. Which do you like best? Cantor's "diagonal" argument is perhaps the most celebrated, but it's not my favorite. De gustibus non est disputandum. I only know Cantor's diagonal argument. What others are the...
by Torn Apart By Dingos
Mon Sep 15, 2008 9:03 pm UTC
Forum: Mathematics
Topic: Bodybuilding forum + recurring decimal =
Replies: 24
Views: 4357

Re: Bodybuilding forum + recurring decimal =

If you had a plane trying to take off from a treadmill moving at the exact speed of the wheels in the opposite direction, and you let a variable x equal 1 if the plane took off or 0.9 repeating if it didn't take off, and then divide x by zero, how long would a thread discussing such an event go bef...
by Torn Apart By Dingos
Mon Sep 15, 2008 9:27 am UTC
Forum: Mathematics
Topic: Randomly Generated Images
Replies: 26
Views: 4394

Re: Randomly Generated Images

I would guess that, asymptotically, 0% of randomly generated pictures make sense, meaning that the percentage of meaningful pictures go to 0 as the resolution of the image increases. My reason for this is that an image, represented as a function from the plane (x,y) to color values (r,g,b), should b...
by Torn Apart By Dingos
Sun Sep 14, 2008 3:30 pm UTC
Forum: Mathematics
Topic: ≡ sign
Replies: 24
Views: 3144

Re: ≡ sign

If "[imath]f(x)\equiv 0[/imath]" formally means [imath]\forall x: f(x)=0[/imath], "[imath]f(x)\not\equiv0[/imath]" might as well mean [imath]\forall x: \neg(f(x)=0)[/imath], i.e. f(x) is never zero.
by Torn Apart By Dingos
Sun Sep 14, 2008 10:51 am UTC
Forum: Mathematics
Topic: ≡ sign
Replies: 24
Views: 3144

Re: ≡ sign

If f(x)=0 means it's true for that particular x, and f(x)\equiv0 means it's true for all x, then what should f(x)\neq0 and f(x)\not \equiv0 mean? Depending on where the negation is placed, it could mean it's never true, not true for all x, or not true for that particu...
by Torn Apart By Dingos
Sun Sep 07, 2008 8:07 am UTC
Forum: Mathematics
Topic: Balls in balls in R^n
Replies: 25
Views: 4139

Re: Balls in balls in R^n

Given an example in n dimensions, take the 3 centres of the small balls, plus the centre of the large ball. These 4 points define a 3-space. Intersect that 3-space with the example and you should get an example in 3 dimensions. (The 3 dimensional balls have the correct radius by choice of the space...
by Torn Apart By Dingos
Tue Sep 02, 2008 10:56 pm UTC
Forum: Mathematics
Topic: Problem 9
Replies: 15
Views: 2238

Re: Problem 9

a=v/t and v=x/t are not (in general) true. a=dv/dt and v=dx/dt, i.e. you get the velocity v by differentiating x once with respect to t, and you get the acceleration a by differentiating x twice with respect to t. I had considered that but because this question isn't about rate of change but about ...
by Torn Apart By Dingos
Tue Sep 02, 2008 10:33 pm UTC
Forum: Mathematics
Topic: Problem 9
Replies: 15
Views: 2238

Re: Problem 9

a=v/t and v=x/t are not (in general) true. a=dv/dt and v=dx/dt, i.e. you get the velocity v by differentiating x once with respect to t, and you get the acceleration a by differentiating x twice with respect to t.
by Torn Apart By Dingos
Fri Aug 29, 2008 9:54 am UTC
Forum: Mathematics
Topic: Favorite mental math tricks/shortcuts
Replies: 61
Views: 9775

Re: Favorite mental math tricks/shortcuts

Klotz wrote:I'm a fan of 2^10=10^3

Also 10^6=2^20, 10^9=2^30, and so on. Pretty convenient.
by Torn Apart By Dingos
Fri Aug 22, 2008 8:20 pm UTC
Forum: Mathematics
Topic: Probability Problem with a Confusing Condition
Replies: 2
Views: 888

Re: Probability Problem with a Confusing Condition

There are 56 different outcomes of throwing three dice and then ordering them: [111,112,113,114,115,116,122,123,124,125,126,133,134,135,136,144,145,146,155,156 ,166,222,223,224,225,226,233,234,235,236,244,245,246,255,256,266,333,334,335,336 ,344,345,346,355,356,366,444,445,446,455,456,466,555,556,56...
by Torn Apart By Dingos
Thu Aug 14, 2008 10:17 am UTC
Forum: Mathematics
Topic: More about xkcd number and other big numbers
Replies: 92
Views: 17655

Re: More about xkcd number and other big numbers

Okay. Am I just using the wrong terminology? Is there a name for the standard model of ZFC (if there is one such - the one where N consists of the elements 0,1,2,... and no others)? Basically, when I do everyday math, say combinatorics or functional analysis, what's the name of the system I am in th...
by Torn Apart By Dingos
Thu Aug 14, 2008 8:30 am UTC
Forum: Mathematics
Topic: Something I've been wondering since 7th grade
Replies: 22
Views: 4433

Re: Something I've been wondering since 7th grade

I don't think we can define a x$y that has any of the nice properties you listed, for the main reason that you can't make the property x+y=x$x$..$x (with y repetitions) make sense when y=1. Right. y+1 repetitions then. So x=x+0, x$x=x+1, x$x$x=x+2, and so on. EDIT: Let x$x:=x+1. Then addition is re...
by Torn Apart By Dingos
Thu Aug 14, 2008 7:57 am UTC
Forum: Mathematics
Topic: More about xkcd number and other big numbers
Replies: 92
Views: 17655

Re: More about xkcd number and other big numbers

I'm referring to this: Assume M can be proven in ZFC to halt in m steps. Then it halts in m steps in all models of ZFC. Assume M can be proven in ZFC not to halt in m steps. Then it cannot halt in any model. Assume M can neither be proven to halt nor proven to not halt (proofs in ZFC). Then it can't...

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