@JasperLoy this view is somewhat accurate for other forms of media that are consumed on the basis of interest. there is a dynamic force at work: if a piece of art is obscure, piracy can lend it the attention it needs to increase its sales, but if it's popular and mainstream, piracy can limit the money that would otherwise be made off of it (although the offset in revenue is only a small fraction of the piracy taking place; most that pirate wouldn't buy if they couldn't pirate).
this works for many books in particular, but not textbooks I don't think, because they are not generally obtained on the basis of a person being interested in the content, but rather on the basis that they need the book for a class. and it's no longer a fractional offset in revenue - it's far more proportional to the actual level of piracy, without the benefit of meaningfully changing the popularity levels of textbooks.
@TedShifrin We were talking a while back about nontrivial maps between real projective spaces; I never made any progress on the complex case, so I posted it as a question.
Well, I know it is repetitive.I have read the proof from different textbooks.But sometimes I feel doubtful about it all.Every time I try to prove it for myself, I fail at some points.I'm asking those guys who are good at geometry, what would you do when you face proving such things?I'm doubtful a...
The past few months I have been thinking about the perfect set of books to study for GRE and quals. I think that list has changed about 10 times. I still have not finalised it yet.
Although I know very little math, I probably know more math books than anyone else, lol.
It's confusing that advanced calculus, mathematical analysis and real analysis can all refer to the same set of topics.
I read that James Stewart built a 20 million dollar house. I wonder how much of that came from the sales of his Calculus.
For example, some people try to prove that the area under a curve is given by the integral without going into the axiomatic definition of area. In such case it is probably better to take that as a definition of area instead.
You guys have an idea on how to prove that a sequence $x_n$ has a subsequence converging to a real number c iff $ \forall \varepsilon>0, |x_n - c| < \varepsilon$ for infinitely many terms n
"for infinitely many n" (not "terms" n, since n isn't a term of the sequence, it's a variable index) is already rigorous. not sure if there's a standard way in symbols to say "there exists infinitely many"
I think in proving that a sequence $ x_n$ has a subsequence converging to a real number c iff $\forall \varepsilon>0, |x_n - c| < \varepsilon $ for infinitely many terms n , we are indirectly using something like the LUB axiom ( completness axiom ) or one of its consequence ( like the Nested Interval theorem )
@Chris'ssis Hi !!! :D ya I did it !! :-) .. but the coefficient of $\zeta(2)\zeta(3)$ should be double of what you showed me last night .. the rest is perfect :D :-)
sorry .. the thing is I made silly mistake 3 times while adding up the final results .. I couldn't see my stupidity last night :) .. but in the morning I figured it out .. :))
@Chris'ssis is the $\sum\limits_{n=1}^{\infty} \psi^{(1)}(n)^4$ tough ? :O
@Chris'ssis hehe instead of saying @Huy Hi .. you could have just said Huy =P (although I have an ambiguity about the pronunciation of Huy .. is it pronounced with a stress on 'u' or like 'Hai' ?!)
@Chris'ssis $\sum\limits_{n=1}^{\infty} \psi^{(1)}(n)^4$ seems very difficult .. it requires evaluation of double harmonic sums of the kind $\sum \frac{H_n}{n^s}H_n^{(r)}$ and $\sum \frac{H_n^{(p)}}{n^q}$ for $p,q,r,s$ varying between $3$ to $6$ .. I don't have elementary solutions for such sums :( $\sum\limits_{n=1}^{\infty} \psi^{(1)}(n)^3$ I could manage because all $p,q,r,s$ were $1,2,3$ or $4$ (I have elementary solutions for them) :| ..
@Chris'ssis not one that I can identify at least :( .. I wonder if its something that I might have done/said that might have ticked him off .. :| Heaven knows what :( ..
Bah! Last night I spent a long time proving that the volume of a regular $n$-simplex is $$\left(\frac s{\sqrt2}\right)^n\frac{\sqrt{n+1}}{n!}$$ only to find out that was not the question asked.
@BalarkaSen Firstly, thanks for looking. Secondly, I feel stupid. What I intended to do was to multiply the denominators and the numerators by $(1-x)$ (effectively one - so as not to change the value of the limit). Let me try that again before I say any more here
okay, let's take $$\lim_{x \to 1} f(x) = \frac{\sqrt{x^2-1}}{x-1}$$. what would be your first step (only) when solving this?
my answer was $\infty$, and that's what the answer in my solution book is (although they got there differently). but wolfram says "two sided limit does not exist". wolframalpha.com/input/…
@topper You're right about $\infty$, that should suffice to prove that the limit doesn't exist. What Wolfram tells is that not only the left and right limit converge, but they aren't equal either.
i.e., if you approach $x \to 1$ from the left then the limit tends to $-i\infty$
You can easily see that by yourself : take $x$ to be something very close to $1$ yet $x < 1$, e.g., $x = 0.999$ (say). Then $x^2 - 1 \leq 0$ but then square root of some negative number is imaginary.
@BalarkaSen Got it. In good news, after learning about limits weeks ago, learning a whole lot of other material in between, forgetting about limits, and now exercising them for the first time, I have gotten my last two questions right! Next...
I don't know how OT we get here but I'm reminded of Alan Shearer the footballer talking on Match of the Day one week, saying that nothing breeds confidence like winning...
Hi people, if $\int_{\Bbb{R}^n} mathbf{x}f(mathbf{x}) \mathrm{d}mathbf{x}$ is equal to the mean vector of a multivariate Gaussian distribution, then what is $\int_{\Bbb{R}^n} \|mathbf{x}\|^2f(mathbf{x}) \mathrm{d}mathbf{x}$? Thanks a lot!
Stumped by $$\lim_{x \to \infty}\frac{1+3+5+...+(2x-1)}{x+3} - x$$. So far, have thought of getting a lowest common denominator which I think looks like $$\lim_{x \to \infty}\frac{1+3+5+...+(2x-1)-x^2-3x}{x+3} $$. But then it gets indeterminate. Grateful for any first steps to solving this, as I don't think mine were very useful
Hi, I was wondering if all elements of $\mathbb{R}$ are limit points of $\mathbb{R}$? It appears to me that it is true, because for any point $r$, we can always find other points that satisfy $|x-r| < \epsilon$ (due to the fact that $\mathbb{Q}$ and $\mathbb{I}$ are dense in $\mathbb{R}$). But I'm not sure if this counts as a valid proof, and I've tried to search it on Google, but I couldn't find anything.
@JasperLoy of course, thanks. "And what does $x$ mean?" I'm not sure what you mean, $x \in \mathbb{R}$ and $x \neq r$. By $\mathbb{I}$ I mean the set of irrational numbers, what is the typical notation?
I'm using notation of Understanding Analysis by Stephen Abbott
Okay, I had a great patch, now I'm flagging. Why can't I just substitute 2 in $$\lim_{x \to 2}\frac{x^2+5x-14}{x^2-5x+6}$$? Why do I have to factorize it? It's not indeterminate when I substitute - I get $-\frac{10}{10}$
@topper: I've done worse mistakes. I mean seriously retarded mistakes. It's quite amazing how long a person can go without realizing something is a mistake.\
I have a pretty simple straightforward question.
Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$
Instinctively, I do the quickest thing I know how to do.
$$\times10!\implies x = \frac{10!}{8!} + \frac{10!}{9!} = 9 \times 10 + 9 = 99$$
That seemed p...
@robjohn Well, your way is very nice. I mean the Cauchy product offers a solution without pen and paper, very easily. Just consider this one $$a_n = b_n = \frac{(-1)^{n}}{n}$$
@JohnJack I need to understand what you are saying here. Are you still talking about convergence in the set of real numbers or just convergence in the set of numbers determined by the sequence? In the former case, the sequence converges to 0. In the latter case, the sequence does not converge.
@Jasper yeah,...say you asked "Give ab example of a real sequence {x_n} for which inf x_n exists but which has no convergent subsequence" then would you always consider that the convergence in with respect to the set of real numbers?
@JohnJack It depends on the context. I would need to see the entire book etc to know. But if you are talking about the real numbers, then (1/n) converges in the real numbers to 0 and it is Cauchy of course.
Let $p_0\leqslant p\leqslant p_1$. If $f\in L^p(X)$, then we can easily see that $f\chi_{\{|f|\geqslant 1\}}\in L^{p_0}(X)$. I'm having a hard time seeing why if $(f_n)$ is a sequence in $L^p(X)$ which converges to $f$, then $(f_n\chi_{\{|f|\geqslant 1\}})$ converges to $f\chi_{\{|f|\geqslant 1\}}$ in $L^{p_0}$. Can anyone help me?
@Ted If you recall our question, Jason DeVito answered it in the positive: there are indeed nontrivial maps $\Bbb{CP}^n \rightarrow \Bbb{CP}^m$ for $n>m$ for any choice of $m$; and it is likely that what $n$ we can choose depends wildly on the homotopy groups of spheres, so not much can be said. One interesting question remains: does $\Bbb{CP}^2 \rightarrow \Bbb{CP}^1$ admit any interesting maps?