## Search found 1179 matches

- Sun May 23, 2010 11:42 pm UTC
- Forum: Mathematics
- Topic: Cardinality of uncomputable numbers
- Replies:
**48** - Views:
**4997**

### Re: Cardinality of uncomputable numbers

That is, we could define some set of "base uncomputables", which could then be combined to form and uncomputable number we like. And I'm pretty sure this set would be infinite. I don't see any reasonable way to do this so that the "base" numbers are "independent" in an...

- Sun May 23, 2010 12:34 am UTC
- Forum: Mathematics
- Topic: Cardinality of uncomputable numbers
- Replies:
**48** - Views:
**4997**

### Re: Cardinality of uncomputable numbers

if we take any one uncomputable number, then we can add any real number and get a new uncomputable number. Uncomputable numbers are also real numbers. In particular, the negative of an uncomputable number is another uncomputable (and real) number. linearly independent units which cannot be reduced,...

- Fri May 21, 2010 5:57 am UTC
- Forum: Mathematics
- Topic: Having trouble understanding Topology
- Replies:
**6** - Views:
**2213**

### Re: Having trouble understanding Topology

The "practical" answer is that there are a lot of interesting topological spaces which don't look anything like the topological spaces you're used to, so you shouldn't rely on your intuition from "familiar" spaces such as the real numbers. The weird spaces are important, too. In ...

- Wed May 19, 2010 3:36 am UTC
- Forum: Mathematics
- Topic: Ring homomorphism question
- Replies:
**7** - Views:
**709**

### Re: Ring homomorphism question

A ring homomorphism sends the identity to the identity. So...

- Tue May 18, 2010 11:59 pm UTC
- Forum: Mathematics
- Topic: Ring homomorphism question
- Replies:
**7** - Views:
**709**

### Re: Ring homomorphism question

The statement is true. Think about where you can send X.

- Thu May 13, 2010 3:58 am UTC
- Forum: Mathematics
- Topic: Question About the Pythagorean Theorem
- Replies:
**40** - Views:
**6034**

### Re: Question About the Pythagorean Theorem

The "point" of the Pythagorean theorem is that the definition of distance is invariant under rotation. From the modern perspective, rotation is actually the more fundamental concept, and distance (and the Pythagorean theorem) arises naturally from it, rather than the other way around. Any ...

- Thu May 13, 2010 3:47 am UTC
- Forum: Mathematics
- Topic: Hyperbolic functions
- Replies:
**9** - Views:
**1814**

### Re: Hyperbolic functions

Does this give any easy-to-calculate arguments for sinh or cosh (like π/6, π/4, π/3, etc are for sin and cos)? No. There shouldn't be any, looking at the expression in terms of exponentials. The nice values of sine and cosine come from the fact that the circle group has plenty of elements of finite...

- Thu May 13, 2010 12:55 am UTC
- Forum: Mathematics
- Topic: Convolution
- Replies:
**19** - Views:
**2335**

### Re: Convolution

The way I think about convolution is that it's the same thing as multiplying the Fourier transforms together. In other words, the amplitude of the first signal at a particular frequency is multiplied by the amplitude of the second signal at the same frequency to get the amplitude of the resulting si...

- Thu May 13, 2010 12:18 am UTC
- Forum: Mathematics
- Topic: Hyperbolic functions
- Replies:
**9** - Views:
**1814**

### Re: Hyperbolic functions

As far as I know, Cleverbeans' is more or less the original definition. It's almost exactly the same as a definition of the ordinary trig functions in terms of the unit circle except that one sign is switched.

- Wed May 12, 2010 7:45 pm UTC
- Forum: Mathematics
- Topic: Small calculus question - squared delta function
- Replies:
**21** - Views:
**8405**

### Re: Small calculus question - squared delta function

PM 2Ring wrote:To my mind, the existance of a well-defined squared Dirac Delta would imply the existence of an inverse Dirac Delta

There's no reason this should be true.

- Wed May 12, 2010 1:35 am UTC
- Forum: Mathematics
- Topic: Small calculus question - squared delta function
- Replies:
**21** - Views:
**8405**

### Re: Small calculus question - squared delta function

Delta isn't in F(R, R). You can read about how the standard approach works here.

- Tue May 11, 2010 8:30 pm UTC
- Forum: Mathematics
- Topic: Small calculus question - squared delta function
- Replies:
**21** - Views:
**8405**

### Re: Small calculus question - squared delta function

That is not the definition of delta. That is the definition of integration against delta. And yes, as others have said, in the standard mathematical formalism for understanding the Dirac (not Kronecker) delta function, its square does not exist in any reasonable sense.

- Mon May 10, 2010 1:03 am UTC
- Forum: Mathematics
- Topic: Can we truly prove anything?
- Replies:
**86** - Views:
**9128**

### Re: Can we truly prove anything?

My point is that the axiom of choice is the only axiom to receive such special treatment (unless you're studying set theory or mathematical logic maybe). The Boolean prime ideal theorem and the Hahn-Banach theorem are of interest to plenty of people who don't study set theory or logic, and they're ...

- Sun May 09, 2010 5:50 pm UTC
- Forum: Mathematics
- Topic: Infinite nines equal what?
- Replies:
**13** - Views:
**1731**

### Re: Infinite nines equal what? (p-adics)

In particular I don't understand why p-adics have to be in a prime base. Even the Wikipedia article shows some examples in base 10, so it seems that composite bases are merely "deprecated" and don't really break anything. The p-adics for p composite are not integral domains. This means th...

- Sun May 09, 2010 3:23 am UTC
- Forum: Mathematics
- Topic: Can we truly prove anything?
- Replies:
**86** - Views:
**9128**

### Re: Can we truly prove anything?

It depends strongly on the field. Mathematicians in various parts of analysis or topology use the axiom of choice largely without comment, since it's implicit in many of the most useful theorems in the field.

- Sun May 09, 2010 3:21 am UTC
- Forum: Mathematics
- Topic: Infinite nines equal what?
- Replies:
**13** - Views:
**1731**

### Re: Infinite nines equal what? (p-adics)

I'm sure this has been said before, but if you use the formula for summing an infinite geometric series to sum a divergent series, you get exactly that answer. This has nothing to do with p-adic numbers. It has everything to do with p-adic numbers. The reason the geometric series formula gives the ...

- Sat May 08, 2010 9:41 pm UTC
- Forum: Mathematics
- Topic: Infinite nines equal what?
- Replies:
**13** - Views:
**1731**

### Re: Infinite nines equal what?

First of all, technically speaking, we probably wouldn't say ...999999 is -1 in any p-adic system because 10 isn't prime. Why not? The 10-adics form a perfectly valid ring; in fact, they're the direct product of the 2-adics and the 5-adics. (They just don't happen to be an integral domain. It might...

- Fri May 07, 2010 2:52 am UTC
- Forum: Mathematics
- Topic: Can we truly prove anything?
- Replies:
**86** - Views:
**9128**

### Re: Can we truly prove anything?

I believe Godel once proved that it is impossible for a logical system to prove it's own consistency. This is not really what the Incompleteness Theorem says. doesn't that mean there is a possibility that they, in fact, aren't. Yep. It is possible that tomorrow someone could show that ZFC is incons...

- Wed May 05, 2010 7:49 pm UTC
- Forum: Mathematics
- Topic: Largest [real] number, and smallest number greater than zero
- Replies:
**11** - Views:
**2588**

### Re: Largest [real] number, and smallest number greater than

Summing the entire series of any unbound set has no answer, but conceptually the sentence sort of made sense to me. Maybe I should quit while I'm ahead :P You probably know that it is possible to define the sum of a countable number of positive real numbers under certain conditions. It is never pos...

- Wed May 05, 2010 6:12 pm UTC
- Forum: Mathematics
- Topic: Largest [real] number, and smallest number greater than zero
- Replies:
**11** - Views:
**2588**

### Re: Largest [real] number, and smallest number greater than

There is no largest real number, and there is no smallest positive real number. The real numbers have a very precise mathematical definition, and both of these properties follow from that definition. The original question is really ill-defined. The problem is the open-ended nature of the word "...

- Tue May 04, 2010 7:13 pm UTC
- Forum: Mathematics
- Topic: A question about sum of sequences
- Replies:
**7** - Views:
**1089**

- Tue May 04, 2010 12:35 am UTC
- Forum: Mathematics
- Topic: I think i broke calculus
- Replies:
**6** - Views:
**983**

### Re: I think i broke calculus

Calculus is more durable than you think.

- Sun May 02, 2010 5:54 am UTC
- Forum: Mathematics
- Topic: Top Colleges for Undergraduate Mathematics
- Replies:
**30** - Views:
**8129**

### Re: Top Colleges for Undergraduate Mathematics

Huh. You don't have to take entrance exams if you're coming abroad from MIT! I feel like I lucked out.

- Sat May 01, 2010 6:16 am UTC
- Forum: Mathematics
- Topic: Top Colleges for Undergraduate Mathematics
- Replies:
**30** - Views:
**8129**

### Re: Top Colleges for Undergraduate Mathematics

Math is one of the areas where the colleges everyone talks about really are (some of) the best. MIT, Harvard, Princeton, and Stanford, for example, all have outstanding math departments. I can't say I know much about universities outside of the US, but I'm studying abroad at Cambridge next year, and...

- Sat May 01, 2010 6:12 am UTC
- Forum: Mathematics
- Topic: Central Binomial Coefficient approximations
- Replies:
**2** - Views:
**1086**

### Re: Central Binomial Coefficient approximations

No. There is a generalization of Stirling's formula which gives approximations of arbitrarily good order which should reproduce these results; see, for example, the Wikipedia article . (People know a lot about asymptotic analysis. If you're interested, you might want to read Flajolet and Sedgewick's...

- Fri Apr 30, 2010 9:44 pm UTC
- Forum: Mathematics
- Topic: Particular type of prime....
- Replies:
**11** - Views:
**1350**

### Re: Particular type of prime....

I don't think the claim was made that it was about abstract groups, although I think I could make it about finite fields without too much trouble. Not canonically. The question is about the multiplicative group of the integers mod p, and elements of the integers mod p don't come with a preferred ch...

- Fri Apr 30, 2010 5:13 pm UTC
- Forum: Mathematics
- Topic: Particular type of prime....
- Replies:
**11** - Views:
**1350**

### Re: Particular type of prime....

Nitpick: this is not a question about an abstract group. As you've stated it, it depends on a particular choice of representatives of congruence classes.

- Fri Apr 30, 2010 2:20 pm UTC
- Forum: Mathematics
- Topic: Continuity of a derivative
- Replies:
**7** - Views:
**2482**

### Re: Continuity of a derivative

And there's a reason why a counterexample is hard to imagine: http://en.wikipedia.org/wiki/Darboux%27 ... nalysis%29

- Thu Apr 29, 2010 4:45 am UTC
- Forum: Mathematics
- Topic: Function Question
- Replies:
**7** - Views:
**693**

### Re: Function Question

Barring some sign technicalities, you can convert this to Cauchy's functional equation. A lot is known about the "weird" solutions to this.

- Wed Apr 28, 2010 9:53 pm UTC
- Forum: Mathematics
- Topic: Natural log as a limit?
- Replies:
**13** - Views:
**1404**

### Re: Natural log as a limit?

Er, sorry, that was imprecise. Yes, I meant "locally monotonic," e.g. in a neighborhood of a point.

- Wed Apr 28, 2010 8:44 pm UTC
- Forum: Mathematics
- Topic: Natural log as a limit?
- Replies:
**13** - Views:
**1404**

### Re: Natural log as a limit?

Right. One can conclude that the answer must be a multiple of the logarithm from pretty much any kind of regularity hypothesis: continuity at a point, monotonicity at a point, differentiability at a point... the counterexamples are truly bizarre; in particular, their graphs are dense in the plane.

- Tue Apr 27, 2010 11:16 pm UTC
- Forum: Mathematics
- Topic: Natural log as a limit?
- Replies:
**13** - Views:
**1404**

### Re: Natural log as a limit?

b^h - 1 = e^{h \log b} - 1 = h \log b + O(h^2) by Taylor expansion (or equivalently, l'Hopital's rule), but this argument can be circular depending on what you've already proven about exponentials and logarithms. Alternately, you might have fun trying to prove that \lim_{h \to 0} \frac{(...

- Tue Apr 27, 2010 5:50 pm UTC
- Forum: Mathematics
- Topic: Rubik's Math
- Replies:
**9** - Views:
**1226**

### Re: Rubik's Math

Moves can be used to bring pieces you cannot see into the two faces that you can see. Moves can be used to rotate all of the other faces to any face that you can see! This seems like a trivial interpretation of the question. My reading of the question is, "if you put a scrambled cube into a gl...

- Mon Apr 26, 2010 4:58 am UTC
- Forum: Mathematics
- Topic: Continuity of this function
- Replies:
**9** - Views:
**1542**

### Re: Continuity of this function

Thomae's function is even Riemann integrable, and its Riemann integral is zero on every interval. The lower Darboux sums are always zero, so it suffices to show that the upper Darboux sums are also always zero. To prove this one uses the fact that for every \epsilon > 0 there are only finitely many ...

- Sun Apr 25, 2010 5:20 am UTC
- Forum: Mathematics
- Topic: About two sums of products of binomial coefficients
- Replies:
**11** - Views:
**1453**

### Re: About two sums of products of binomial coefficients

My understanding is that the combinatorial proof of the second identity is hard; I don't actually know it. The combinatorial proof of the first one is probably not easy either, since it's closely related.

- Sat Apr 24, 2010 5:58 pm UTC
- Forum: Mathematics
- Topic: About two sums of products of binomial coefficients
- Replies:
**11** - Views:
**1453**

### Re: About two sums of products of binomial coefficients

Both of these have very short proofs using generating functions (which are also covered in GKP). Is that "constructive" enough for you?

- Fri Apr 23, 2010 6:33 pm UTC
- Forum: Mathematics
- Topic: Where am I on the curve?
- Replies:
**33** - Views:
**3052**

### Re: Where am I on the curve?

when I see certain terms such as the epsilon-delta kind of definition, I become a bit worried about where I am. I wonder how I'll do in Calculus Unless you're actually being graded on a curve, it shouldn't matter how you compare to other students your age. As long as you're interested in and motiva...

- Fri Apr 23, 2010 5:17 am UTC
- Forum: Mathematics
- Topic: Laurent expansion of cot(z)
- Replies:
**3** - Views:
**3068**

### Re: Laurent expansion of cot(z)

Write cot(z) in terms of e^{iz} and try to relate it to the generating function for the Bernoulli numbers. Are you sure the professor didn't just want you to write down a few terms?

- Thu Apr 22, 2010 5:32 pm UTC
- Forum: Mathematics
- Topic: In non-base 10 mathematics, would 1/3 still be repeating?
- Replies:
**2** - Views:
**828**

### Re: In non-base 10 mathematics, would 1/3 still be repeating

In base 3, it's 0.1. The fractions that repeat in a different base are precisely those such that the denominator has at least one prime factor which doesn't divide the base.

- Thu Apr 22, 2010 3:01 am UTC
- Forum: Mathematics
- Topic: Where am I on the curve?
- Replies:
**33** - Views:
**3052**

### Re: Where am I on the curve?

Why does it matter?