## Search found 146 matches

Fri May 22, 2009 1:03 am UTC
Forum: Mathematics
Topic: ITT: Why Math Is Awesome.
Replies: 51
Views: 6195

### Re: ITT: Why Math Is Awesome.

My favorite part of math is it's infinite difficulty and complexity. You start out learning things that most people don't know, then you learn things that only interest mathematicians, then things which only interest specialists, and pretty soon you're thinking about problems that humans probably wo...
Tue May 05, 2009 7:36 pm UTC
Forum: Mathematics
Topic: Boundedness vs. total boundedness
Replies: 5
Views: 3004

### Re: Boundedness vs. total boundedness

I really don't think it has to do with dimension, because dimension is usually a local property, and this is not. For example, R n with the metric min(|x-y|, 1) has exactly the same local metric properties, including dimension, as standard R n . The property in question is much more "large scal...
Wed Apr 29, 2009 3:33 am UTC
Forum: Mathematics
Topic: Sierpenski
Replies: 6
Views: 1118

### Re: Sierpenski

Pascal's triangle mod 2.
Fri Mar 13, 2009 3:26 am UTC
Forum: Science
Topic: Evolution of another intelligent species on Earth?
Replies: 42
Views: 4555

### Re: Evolution of another intelligent species on Earth?

Cycle, I think you should read some more biology. And the thread. Is there something wrong with my post that I can't see? Maybe I'm thinking of a much more narrow form of intelligence than you are? I'm talking about self-conscious, capable of philosophy (-ish) smarts. As far as biology goes, the on...
Fri Mar 13, 2009 3:02 am UTC
Forum: Mathematics
Replies: 367
Views: 83494

Hey guys I was developing a theory about 1/3 in Z 6 . I've tried calling it "infinity", but there's still a few issues. 2 = (2 * 3) * 1/3 = 0 * 1/3 = 0!!! It must be because of the strange nature of infinity. Or it's a new theory that mathematicians will develop when they understand Z 6 be...
Thu Mar 12, 2009 8:32 pm UTC
Forum: Science
Topic: Evolution of another intelligent species on Earth?
Replies: 42
Views: 4555

### Re: Evolution of another intelligent species on Earth?

On the original topic, I think another species developing intelligence while humans are still around is pretty unlikely. First of all, it would have to be something large. An insect could never have the head volume, or the necessary calories to run an intelligent brain. Secondly, an intelligent brai...
Wed Mar 11, 2009 6:59 am UTC
Forum: Mathematics
Topic: Definitions & explanations in English/Actual uses for math
Replies: 19
Views: 1854

### Re: Definitions & explanations in English/Actual uses for math

ThomasS wrote:Lockhart's Lament

That's wonderful. Thanks for linking that.
Wed Mar 11, 2009 2:03 am UTC
Forum: Mathematics
Topic: Definitions & explanations in English/Actual uses for math
Replies: 19
Views: 1854

### Re: Definitions & explanations in English/Actual uses for math

Look, math isn't useful. OK, so you can give a big list about how much scientists and engineers use math, but for the average person, it's useless. Calculus won't help you balance a checkbook. Hell, even if you wanted to balance a checkbook in these online banking days, you can buy a calculator for ...
Tue Mar 10, 2009 8:12 pm UTC
Forum: Mathematics
Topic: Graph: F(x)=x^∞
Replies: 25
Views: 3143

### Re: Graph: F(x)=x^∞

For a quick answer: \lim_{x\to\infty}\frac{1}{x^2}=\frac{1}{\infty^2} = \frac{1}{\infty} = 0 \lim_{x\to\frac{\pi}{2}}\frac{1}{cos(x)}=\frac{1}{cos(\frac{\pi}{2})} = \frac{1}{0} = \infty Meaning, in the context of limits at least (an underlying basis of many other things like derivat...
Tue Mar 10, 2009 4:32 pm UTC
Forum: Mathematics
Topic: Summation of consecutive integers.
Replies: 18
Views: 1941

### Re: Summation of consecutive integers.

Ok... then I guess I should clarify that everywhere I said "integer" I really should have said "positive integer" or "natural number". Thanks for the reply though. I'll have time to look at it more closely later. Of course, any positive integer that is the sum of conse...
Tue Mar 10, 2009 4:24 pm UTC
Forum: Mathematics
Topic: Readings on Vectors and Three Dimensional Space
Replies: 11
Views: 1653

### Re: Readings on Vectors and Three Dimensional Space

What you should take away from this is that the cross product doesn't live in quite the same space as its constituent parts. And if you think about it, physicists recognize this distinction as well: why else are the units for torque considered distinct from the units for work? The cross product is ...
Thu Mar 05, 2009 5:04 am UTC
Forum: Mathematics
Topic: Gauss v Euler
Replies: 48
Views: 16435

### Re: Gauss v Euler

problem-solver or a theory-builder Interesting read. And a nice coincidence, because I consider Atiyah to be one of the greatest mathematicians of recent time. Much more than Erdos or Godel. Between Euler and Gauss, I'd have to pick Gauss. Ancient? Euclid beats Archimedes and Pythagoras combined. T...
Wed Mar 04, 2009 6:57 am UTC
Forum: Mathematics
Topic: Augmented matrix
Replies: 12
Views: 1535

### Re: Augmented matrix

Honestly, can you just start deleting these threads? There should be one sticky, called "Help with homework in pre-calc classes". Clear up the clutter on this forum SO FRIGGIN MUCH. Of course, it's a theoretical possibility that no one would ever go into the thread, ever. It's a risk I'm w...
Wed Mar 04, 2009 6:05 am UTC
Forum: Computer Science
Topic: The Great Operating System Disscussion
Replies: 82
Views: 10569

### Re: The Great Operating System Disscussion

I've been using a MacBook Pro as my primary computer for the last couple of years, and while I like it a lot better than my old Windows desktop, I'm forced to say that the next laptop I buy, I am putting Linux (probably Ubuntu, at least for a little while) on. The reasons? Well, I find it kind of a...
Wed Mar 04, 2009 5:56 am UTC
Forum: Mathematics
Topic: Is our mathematics system flawed?
Replies: 57
Views: 5909

### Re: Is our mathematics system flawed?

"X is uncountable" is not a statement you can make in the language. The language has parentheses, 0, 1, +, *,=, variables, quantifiers, and logical symbols. You can't even talk about sets. You can say "there are at least three different numbers" by saying something of the form \...
Tue Mar 03, 2009 10:34 am UTC
Forum: Mathematics
Topic: Is our mathematics system flawed?
Replies: 57
Views: 5909

### Re: Is our mathematics system flawed?

I might be misunderstanding you, but the field axioms can neither prove nor disprove the statement "1+1=0", so they're not complete. On the other hand, the field axioms plus the axiom schema for characteristic zero (i.e. for each n>0, the axiom 1+1+...+1≠0, where n 1s appear in the sum), ...
Tue Mar 03, 2009 10:22 am UTC
Forum: Computer Science
Topic: Solaris?
Replies: 14
Views: 1420

### Re: Solaris?

I'm a big fan of Sun's other products, such as OpenOffice and Virtual Box (and obviously Java). So recently I gave it a shot, virtualized. I can't speak very intelligently about it, but the impression I got was that it had a lot more features than most Linux distros (time sliders been mentioned). It...
Tue Mar 03, 2009 6:32 am UTC
Forum: Mathematics
Topic: Continuum-many Q-linearly indepedent real numbers
Replies: 5
Views: 881

### Re: Continuum-many Q-linearly indepedent real numbers

There's a few standard facts about cardinal arithmetic that'll work. For example, if you know (aleph 0)*A = A for any infinite cardinal A, that would do it. Alternatively, if you prove A n = A for any natural n, you'll be fine. These facts are pretty easy to prove. Alternatively, you could prove the...
Mon Mar 02, 2009 8:17 pm UTC
Forum: Mathematics
Replies: 159
Views: 21673

Some pics of the same thing. http://demonstrations.wolfram.com/TheBanachTarskiParadox/ Intriguing. That's awesome. I wouldn't really call it the same thing as Banach-Tarski, since one paradox is "look how weird non-measurable sets are" and one is "look how weird isometries of the hyp...
Mon Mar 02, 2009 2:33 am UTC
Forum: Mathematics
Topic: Representation theory help
Replies: 23
Views: 2061

### Re: Representation theory help

I don't think it's much easier at all: it's exactly the same proof. But it is time saving: most people are familiar with determinant long before they're familiar with homomorphisms of non-abelian groups. The representation theory makes it so you don't have to develope the same theory twice, I guess.
Mon Mar 02, 2009 1:10 am UTC
Forum: Mathematics
Replies: 159
Views: 21673

The Banach-Tarski paradox has been mentioned a couple of times, but I personally favor the following variant(hopefully, i'm not misremembering; it's been a while since I saw this): The Hyperbolic plane can be broken into a finite number of disjoint regions(regions, i.e. connected sets) not dust-lik...
Sun Mar 01, 2009 11:46 pm UTC
Forum: Mathematics
Topic: Representation theory help
Replies: 23
Views: 2061

### Re: Representation theory help

Here's a nice, simple application: When constructing A n , you first have to construct a map S n -> Z 2 (which is the sign of the permutation), and then define A n to be the kernel. But this map is really hard to define: you have to prove that the sign of a permutation is well defined, which is non-...
Thu Feb 26, 2009 8:45 am UTC
Forum: Mathematics
Topic: Traveling across a square
Replies: 24
Views: 6714

### Re: Traveling across a square

A much better generality I think is something like Lipschitz functions, or absolutely continuous functions. Probably with a metric like \sup_{x \in [0,1]} |f(x)-g(x)| + \limsup_{\epsilon \to 0} \sup_{|x-y|< \epsilon} \frac{|f(x) - g(x)|}{|x-y|}. I say something like t...
Mon Feb 23, 2009 2:53 am UTC
Forum: Mathematics
Topic: Traveling across a square
Replies: 24
Views: 6714

### Re: Traveling across a square

More important is to actually consider the topology of the space of paths. It seems obvious that lim (sawtooth curves) = diagonal, but what do you actually mean by limit? Most people intuitively use the C 0 topology (that is, with the sup norm). But length isn't even defined on C 0 , as many curves ...
Mon Feb 23, 2009 1:51 am UTC
Forum: Religious Wars
Topic: Why are Microsoft evil?
Replies: 180
Views: 44048

### Re: Why are Microsoft evil?

Microsoft's is one of the few companies I know of who make a business practice out of impeding technological advancement. OK, so they're not like Scientology evil or oil company evil, but anyone who doesn't see them as the most evil single entity in technology is provably wrong. Microsoft has a long...
Sun Feb 22, 2009 7:52 pm UTC
Forum: Mathematics
Topic: Straightedge and graph paper geometry
Replies: 25
Views: 2532

### Re: Straightedge and graph paper geometry

Doing things that way, I'm pretty sure you can construct all numbers which are constructable with ruler and compass. This basically says any number which is a constructable length is constructable on the x-axis, right? Suppose we have a constructable length, a. Then the remaining leg a right triangl...
Sat Feb 21, 2009 9:18 pm UTC
Forum: Mathematics
Topic: Isomorphism theorems (in categories I think)
Replies: 14
Views: 2240

### Re: Isomorphism theorems (in categories I think)

So does the theorem still hold for nonabelian groups? My homological algebra is weaker than it should be, but is H^1(G,H) still defined and everything if H isn't abelian? I don't know how to do it, but I'm pretty sure the answer is yes. And I imagine that it's defined in such a way so that H 1 (X, ...
Sat Feb 21, 2009 12:41 am UTC
Forum: Mathematics
Topic: Isomorphism theorems (in categories I think)
Replies: 14
Views: 2240

### Re: Isomorphism theorems (in categories I think)

Erm, the first singular cohomology group of RP^2 is trivial. The homology group is Z/2Z. You seem to be using the two interchangeably. Any map from RP^2 into the circle must necessarily be trivial on fundamental groups, thus lifts to a map to the universal cover of the circle, which is the real lin...
Fri Feb 20, 2009 8:34 pm UTC
Forum: Mathematics
Topic: Isomorphism theorems (in categories I think)
Replies: 14
Views: 2240

### Re: Isomorphism theorems (in categories I think)

There is a faithful functor from the category of groups to the category of topological spaces, which takes a group G to something called an Eilenberg-MacLane space for G or K(G,1). If you look if the category of topological spaces where the maps are defined only up to homotopy, this functor is full...
Fri Feb 20, 2009 6:01 pm UTC
Forum: Mathematics
Replies: 15
Views: 2001

Wait, huh? I thought every field has a unique algebraic closure. Nope. The p-adic rationals have infinitely many non-isomorphic algebraic closures. I'm pretty sure you mean algebraic extensions. The statement is "C contains the algebraic numbers" works, I guess. This is much weaker than t...
Fri Feb 20, 2009 4:45 pm UTC
Forum: Mathematics
Replies: 15
Views: 2001

Wait, huh? I thought every field has a unique algebraic closure. Anyways, "R has a unique algebraic closure" isn't equivalent. Neither is "there exists the algebraic numbers". The statement is "C contains the algebraic numbers" works, I guess. That's why every proof of ...
Wed Feb 18, 2009 10:41 pm UTC
Forum: Mathematics
Topic: Isomorphism theorems (in categories I think)
Replies: 14
Views: 2240

### Re: Isomorphism theorems (in categories I think)

There is a faithful functor from the category of groups to the category of topological spaces, which takes a group G to something called an Eilenberg-MacLane space for G or K(G,1). If you look if the category of topological spaces where the maps are defined only up to homotopy, this functor is full...
Tue Feb 10, 2009 2:51 am UTC
Forum: Mathematics
Topic: Improper Integrals at infinity
Replies: 15
Views: 1745

### Re: Improper Integrals at infinity

bizkut wrote:it's just... strange, and unintuitive and illogical in my mind, which is just against what I've seen of math thus far in my career.

Keep going. Stuff gets way weirder.
Tue Feb 10, 2009 2:04 am UTC
Forum: Mathematics
Topic: Improper Integrals at infinity
Replies: 15
Views: 1745

### Re: Improper Integrals at infinity

You have to remember, math isn't always intuitive, but everything has a precise meaning. When we say \int_1^\infty \frac{1}{x}\,dx = \infty , this really means that the function f(x)=\int_1^x \frac{1}{u}\,du approaches infinity. The integral gets as large as we like, if we're willing to take...
Sun Feb 08, 2009 6:31 am UTC
Forum: Mathematics
Topic: The probability of impossible
Replies: 63
Views: 7229

### Re: The probability of impossible

Or, for a more trivial example, consider the state space {0,1} with probabilities P({1}) = 1, P({0})=0. Then 0 is a possible event. Unless "possible" means something other than "a point in the event space"?
Sat Feb 07, 2009 11:58 pm UTC
Forum: Mathematics
Topic: taylor expansion of sin(x)
Replies: 15
Views: 3306

### Re: taylor expansion of sin(x)

Well, the obvious complex solutions are e^{\omega z} , where omega is an nth root of unity. But if \omega = a + bi , we can say that the complex functions e^{az} \cos(bz) and e^{az} \sin(bz) must satisfy the same properties (since they span the same space as e^{\omega z}, e^{\bar{\om...
Fri Feb 06, 2009 11:44 pm UTC
Forum: Mathematics
Topic: taylor expansion of sin(x)
Replies: 15
Views: 3306

### Re: taylor expansion of sin(x)

In fact, all functions with periodic derivatives are expressible with just e^x and trig functions, not just those of order four. For example, the function exp(x/2)cos(x√3/2) is equal to it's own third derivative.
Fri Feb 06, 2009 9:52 am UTC
Forum: Logic Puzzles
Topic: Hilbert's Hotel
Replies: 102
Views: 10411

### Re: Here's a fun problem.

I think the cardinality of the set of bijections between the two sets is equal to the cardinality of the set of reals (by a Cantor-esque argument) You don't even need a diagonal argument. If F is the set of all such functions, then F is a subset of P(NxNxN), so |F| \leq c. But any function determin...
Fri Feb 06, 2009 7:21 am UTC
Forum: Mathematics
Topic: Number Game!
Replies: 64
Views: 4262

### Re: Number Game!

There's a way to do what you were trying to do. I think I saw it on rec.puzzles: N = -\frac{\log \left(\frac{\log \left(\sqrt{\sqrt{...\sqrt{\sqrt{4}}}}\right)}{\log 4}\right)}{\log{\sqrt{4}}} The ellipsis means there are N nested square-root signs within that inner parentheses. Thu...
Sun Feb 01, 2009 6:35 am UTC
Forum: Mathematics
Topic: Life and math
Replies: 21
Views: 2335

### Re: Life and math

Based on the OP, you're not trying to actually explain interesting math related to the Game of Life. You're trying to make the simple parts of it sound more mathematical, in order to confuse or impress people? Yeah, that should be doable. diotimajsh gave some good suggestions, like using {0, 1} ins...