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1d
comment Finding math research problems
There has to be a reason you're interested in math - so do that! You don't necessarily have to wait for somebody to hand you an open problem. There's something in the mathematical universe that interests you. So hone in on that, focus your interest, and if you can't find a problem from somebody else, you can always make up your own.
1d
comment Of strings and substrings: A problem of probability
The "Goulden-Jackson cluster method" is a sort of calculus for counting words according to occurrences of subwords that might be useful. See my answer here.
Jul
27
answered Find the total number of matchings in a complete graph with even vertices
Jul
25
comment An approach to approximating the harmonic series.
I'm not sure what you mean. You might be interested in the digamma function $\psi(x)$ that satisfies $\psi(x+1) - \psi(x) = 1/x$.
Jul
25
comment An approach to approximating the harmonic series.
See: en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant :)
Jul
22
comment Article writing: How to represent a matrix by its elements?
I would use $a_{ij}$ or $a_{i,j}$. The dimensions should be pretty clear from the context, although your notation would be fine too.
Jul
17
comment Online publication of short essays
Sounds like a good blog post to me.
Jul
14
comment On the value of proofs vs counterexamples
You could disprove a statement that way, but ultimately any proof is logically equivalent to showing that there exists a counterexample, since that's what it means for such a statement to be false. So either you can trace through the logic of the proof and discover that you essentially found a counterexample in disguise, or else it's really a nonconstructive proof.
Jul
14
comment Notation for “set of leaves of a tree” when leaves are “repeated”
I would just say each leaf is labeled with any arbitrary number rather than saying that the leaves are repeated, although you might be able to get away with that if the context is clear.
Jul
14
comment On the value of proofs vs counterexamples
Yes, a proof of such a statement has to prove that a counterexample exists. However, this may be a nonconstructive proof, meaning that it proves counterexamples exist without pinpointing a specific counterexample. Alex's question below gives an example.
Jul
4
comment GRE Math subject test prep
If I recall the Princeton Review one is okay. I think Agarwal & Elsner has some mistakes but is still fairly useful. I could be mixing these up, though, as it has been a while since I've looked at them and the newer editions may be better. Also, when I registered for the GRE they sent me a CD with at least one practice exam on it. Not sure if they still do that.
Jul
3
comment open problems regarding functions
It would help if you were a lot more specific about what kind of functions you are interested in.
Jul
3
comment open problems regarding functions
This is a little bit vague, since any problem can be phrased in terms of a function. e.g., is there a function $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $\lim_{x \rightarrow \infty} f(x) = \infty$ and so that $f(x)$ is less than the size of the set $\{ n < x: n \text{ is a twin prime}\}$? This is equivalent to the twin prime conjecture...
Jul
3
comment Is there a name for the function of a semicircle?
I don't think there's a special name for this. It's fine to just label it $S(x)$ or whatever.
Jul
2
comment values of sin of multiples of 10?
I would also guess that your professor means $\sin(1\pi), \sin(10\pi)$, etc. In my experience, it's very common for students to confuse the $\pi$ in expressions such as $\sin (3 \pi)$ for a unit like in,cm etc. rather than the number $\pi = 3.14159...$
Jun
29
comment Any shortcut to calculate factorial of a number (Without calculator or n to 1)?
You could calculate $n! = \exp(\log(1) + \log(2) + \ldots + \log(n))$ using a log table.
Jun
26
comment Proof: Derivative of $(-1)^{x}$
The expression $(-1)^x$ is not defined in general. You can interpret it to mean $e^{ \pi x i}$, and the formula follows from that. But you could just as easily assume that it means $e^{3 \pi x i}$ which has the derivative $3 \pi ie^{3 \pi x i}$.
Jun
25
comment Linear extension of a set
I think what Miriam wants is to see all of the ordered pairs of the relation given by the linear extension written out explicitly.
Jun
25
comment What does it actually mean by a “Characteristic Polynomial”?
@Ian: The characteristic polynomial and its roots go hand in hand, so I don't see how one could be more significant than the other. In any case, the coefficients of the characteristic polynomial can be useful as well, especially for the Laplacian of a graph where they give the number of spanning forests (although this is too far afield to be an answer to the OP.)
Jun
25
comment What does it actually mean by a “Characteristic Polynomial”?
The characteristic polynomial of $M$ is $p(x) = \det(Ix - M)$. It's only defined for matrices, although there are numerous other unrelated things also called the "characteristic polynomial" in different contexts.