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by t0rajir0u
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...
by t0rajir0u
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,...
by t0rajir0u
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 ...
by t0rajir0u
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...
by t0rajir0u
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.
by t0rajir0u
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 ...
by t0rajir0u
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...
by t0rajir0u
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...
by t0rajir0u
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.
by t0rajir0u
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.
by t0rajir0u
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.
by t0rajir0u
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.
by t0rajir0u
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 ...
by t0rajir0u
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...
by t0rajir0u
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.
by t0rajir0u
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 ...
by t0rajir0u
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...
by t0rajir0u
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...
by t0rajir0u
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...
by t0rajir0u
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 "...
by t0rajir0u
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.
by t0rajir0u
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.
by t0rajir0u
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...
by t0rajir0u
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...
by t0rajir0u
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...
by t0rajir0u
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.
by t0rajir0u
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
by t0rajir0u
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.
by t0rajir0u
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.
by t0rajir0u
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.
by t0rajir0u
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{(...
by t0rajir0u
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...
by t0rajir0u
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 ...
by t0rajir0u
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.
by t0rajir0u
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?
by t0rajir0u
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...
by t0rajir0u
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?
by t0rajir0u
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.
by t0rajir0u
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?

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