Search found 1179 matches

Sun May 23, 2010 11:42 pm UTC
Forum: Mathematics
Topic: Cardinality of uncomputable numbers
Replies: 48
Views: 6106

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: 6106

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: 2830

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: 868

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: 868

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: 7690

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: 2062

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: 2885

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: 2062

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: 9485

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: 9485

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: 9485

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: 11637

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: 2116

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: 11637

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: 2116

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: 2116

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: 11637

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: 3079

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: 3079

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: 1380

Re: A question about sum of sequences

Tue May 04, 2010 12:35 am UTC
Forum: Mathematics
Topic: I think i broke calculus
Replies: 6
Views: 1147

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: 9195

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: 9195

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: 1259

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: 1677

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: 1677

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: 2893

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: 810

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: 1628

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: 1628

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: 1628

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: 1721

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: 1871

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: 1704

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: 1704

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: 3837

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: 3736

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: 937

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: 3837

Re: Where am I on the curve?

Why does it matter?