@SabyasachiMukherjee. Yes. That's right. Let $(X,d_x)$ and $(Y,d_y)$ be metric spaces and $f:X \to Y$. Then $f$ is continuous on $X$ iff $f^{-1}(U)$ is open in $(X,d_x)$ for every open set $U$ in $(Y,d_y)$.
Here's a taste. So I'm sure you've seen many functions. You have worked with these functions, learnt certain characteristics and so on. However, the domain of these functions were more or less - the reals. My question to you is, what happens if I put a complex number into these functions? I.e. - what if the domain is the complex field?
Is sin(z) still bounded between -1 and 1? It's a great subject (y).
And you definitely use that a lot in Complex Analysis. From what I know there are 3 ways to define $sin$. The first is its trigonometric ratio: Opp/Hyp. The second is its transcendental function. And the third is expressing sin in terms of exponentials.
nice weather, good food, and some excellent hill-walking
Ah - but Tenerife is not like africa in many ways - more like a spanish resort just a bit further south - no poisonous snakes/insects or animals that eat people, for instance
@JasperLoy That sounds like some serious reading effort is required. When I was young I once spent a whole summer sitting outdoors in wonderful weather reading a book on Buddhism.
A question I just created. Maybe you like it. Compute $$\lim_{n\to\infty}\int_0^{3/4}\int_0^{15/16}\cdots\int_0^{(4 n^2 - 1)/(4 n^2)}\frac{1}{1-(x_1x_2\cdots x_n)^2} dx_1dx_2\cdots dx_n$$
Let $\Omega$ be a region $\subset \subset \mathbb{R}^{n}$, and suppose that both $u_{\infty}$ and $\varphi$ $\in C_{0}^{\infty}(\Omega)$. Then, I want to perform integration by parts on the following:
$\displaystyle \int_{\Omega}\sum_{j=1}^{n} \frac{\partial u_{\infty}}{\partial x_{j}} \frac{\p...
Tell me one thing @Pedro, lots of people wander around saying that mathematics is useless, do you care to contradict them? So why me? C'mon, ones a stupid always a stupid, you can't do anything about them.
@BalarkaSen Uselessness is completely subjective, but of course math is not, much like engineering isn't useless. Most things have a purpose -- one might be wondering into philosophical speech, though. I don't think you're stupid, but you should be careful about your attitude towards yourself, your learning and your respect to things you don't know.
3
One forms an opinion of something if one studies it. Else, it's just gibberish.
@PedroTamaroff "I don't think you're stupid" That's not a frank truth, c'mon. Anyways, I do, though, and that's a reasonable explanation of how one says something is useless etc. without studying it, no?
For example, many people that are just learning mathematics look for "intuition" (I did) when "intuition" simply means "let's study it enough so it makes sense to me". Someone explained to me, and I agreed, that intuition is something we build on experience. Mathematics happens to defy our usual understanding of things, hence our intuition (...)
In the lower levels, one can try to overlap certain ideas to every-day situations or known things, but then one needs to develop a new intuition, a mathematical intuition, to make sense of things.
@BalarkaSen Well, you seem to be good a learning the big words and using them. I think you should also learn the basics, since the provide with the examples the big guys built modern math over.
@skullpatrol "estrangement"?
Intuition would be like "let's get this into my comfort zone, into a form I can with things I already know."
Sometimes the interesting situation is to leave our comfort zone... I think that's how we progress.
the ability to understand something immediately, without the need for conscious reasoning. "we shall allow our intuition to guide us" synonyms: instinct, intuitiveness
Er I think that may have gotten lost-I'm just gonna repost- Does every sequence in Q also have a monotone subsequence? I know it does in R, and the argument doesn't seem to require anything like LUBP so...
@PedroTamaroff I am doing a limit that gives the number of divisors at the pole of zeta, but is equal to either -1 or Complex infinity for a zeta zero when simplifying it.
> Rather than knowing the correct rules of thought theoretically, one must have them assimilated into one's flesh and blood ready for instant and instinctive use. Therefore, for the schooling of one's powers of thought only the practice of thinking is really useful.
Let $\Omega$ be a region $\subset \subset \mathbb{R}^{n}$, and suppose that both $u_{\infty}$ and $\varphi$ $\in C_{0}^{\infty}(\Omega)$. Then, I want to perform integration by parts on the following:
$\displaystyle \int_{\Omega}\sum_{j=1}^{n} \frac{\partial u_{\infty}}{\partial x_{j}} \frac{\p...
