if you're affiliated with an institution (e.g., are either an undergraduate or graduate student at a university) they may very well have access to this journal
@MikeMiller :P .. it will be the end of my vacation by the time I reach there by foot :P .. I can just wait in that case till the last day and get a flight :P
Oh, well, I'd been thinking about becoming a member for a long time, and I'm now a subscriber to Crux... but they're going to snail mail me my online access...
so I'm sure that in a mere 2 weeks, I'll have access
@LucasZanella Ok, I've done what I can do. I don't actually quite know what will happen. I foresee three possibilities: maybe if you logout and login, your chat name will be correct now; or maybe it will update within 24 hours or so, as many things sort of work like that; or maybe I didn't do anything successfully and you should click the "contact us" link at the bottom of one of the main sites to talk to the real SE mods
oohh, but I'm optimistic, as responding to your post used your name
Suppose $P$ is a permutation that consists of a single cycle, and is not a product of two or more smaller disjoint cycles. Is it normal to say that $P$ is “cyclic”?
Thanks. I see Wikipedia is iffy about it also; it has an entry for cyclic permutation which says it sometimes means what I said, and sometimes means something more general.
Yeah, shame on my university. I also have enjoyed every copy of the Mathematical Gazette that I've read (from the UK), but as far as I can tell, it is not distributed electronically at all
Define $C=\{f:[0,1]\to[0,1]\}$ such that elements of C are continuous. Then define a metric on $C$ of the form $d(f,g)=\max_{x\in[0,1]}|f(x)-g(x)|$. Let $C_i$ and $C_s$ be the set of injective and surjective elements of $C$. Prove or disprove that $C_i$ is closed and $C_s$ is closed.
my question is that how do I approach such questions in general. I know the definitions of open and closed. The way I see it there are two ways. Either I can go about saying $C$ is a subset of larger space of continuous functions and try and show that $C_i=D\cap C$ where $D$ is closed in larger space
Or I directly go into trying and proving that complement of $C_i$ and $C_s$ are open by selecting a point and trying to construct an open ball around it that it lies completely in the complement.
This is a sample question from one of the past exams. My exam is on monday. The issue is that I don't understand how to approach such problems in general spaces (ball approach or set approach) and I do not even know whether I will just be wasting time or not because the thing might be false
I personally always shoot for the ball approach first
@r9m raises a nice point, that sequence lives in that set, but the limit is definitely not in the set so it cannot be sequentially compact $\implies$ its not compact in general so it cannot be closed
@user52932 its a sequence of functions in $C_i \cap C_s$ .. ie bijective (injective and surjective) functions that does not have limit point in $C$ .. so neither $C_i$ or $C_s$ are closed .. I guess
presumably you're not in a topology course, otherwise you would need to take into consideration a couple of other factors (you would need to make sure that it's hausdroff)
yeah I'll be graduating soon and studying math on my own too. I want to do some measure theory and probability theory. I am not that good at proofs though :(
Like I can understand them when they are given in the book and they seem ridiculous
but i lack the creativity to come up with own proofs
I have another general question too. Sometimes on exams you get questions where they give you weird sequences of functions and then they ask you to evaluate limit of the sequence etc...for example consider this sequence
$f_n=\frac{sin(\frac{x}{n})}{x\frac{1}{n}}$ and then question asks what happens to $f_n'$ and $f_n$ as $n$ gets large.
and what would be the limits...
is there a way to do this stuff analytically and directly
@r9m I see. So in general to check connectedness, it is good to check if the set is convex. Becuase that will guarantee path connectedness and hence connectedness (i.e. $C$ cannot be union of two disjoint sets)
@robjohn is it true that $C_0([0,\infty)) = \{f: f \text{is continuous} and \lim\limits_{x \to infty} f(x) = 0\} = \overline{V}$, where $V=\{e^{-x}P(x): P\text{is a polynomial}\}$ ?
so what's the difference between this and the math overflow chat? my first guess is that this room gets more homework questions and drive-by shootings and such
well then, i'm sure i have questions; but none at this moment... it's 4:37 am in georgia, i'm listening to minimal techno, and reading this recent paper on gamma error bounds : arxiv.org/pdf/1310.0166v2.pdf ;)
Inspired by the coverings of compactified modular curves, I was wondering if similar can be done in general, i.e., can galois groups be realized as galois coverings of compact Riemann surfaces?
You showed me that $$ \int_0^{T} f(nx)\,\mathrm{d}x = \int_0^{T} f(x) \,\mathrm{d}x$$
Given that $f(x+T) = f(x)$.
What can one say about the sine case?
$$ \int_0^{T/4} f(\sin n x) \,\mathrm{d}x $$
The $T$ case is clear, but what abut $T/2$ or $T/4$?
It is already know that $$ \int_0^{\pi/2} \log (\sin 2x) \,\mathrm{d}x = \int_0^{\pi/2} \log( \sin x)\,\mathrm{d}x $$ So it must hold in some cases, even when not integrating over a whole period :p
kirill: well, it seems unlikely that euler-mascheroni is algebraic, but by all means -- keep trying. when you run out of meth continued fractions will be waiting for you.
Is $\text{Res}[f(z), \infty]= \text{Res}[f\left(\frac{1}{z}\right), 0]$, assuming that $f(z)$ converges for $|z|>R$ for some $R$? I have an example that seems to indicate not, but I don't know why.
kirill: just curious -- what's the application? is it school, or for software, or simply the joy of discovery? the method, especially in this case, really depends on the precision needed for the application.
And probably I am using wring approach to solve it, may be I don't need to find the root of this equation
@DavidKirby So, I have function f(x)=(x-1)arcsinx on [-1,1], and I need to verify somehow that the zero(s) of f represent points where function has tangent lines
oh no i've already been shot and i almost got shot earlier when that absurd expansion of a step function was posted as an answer to a problem with no explicit answers
from now on i'm just throwing fruit and making topology jokes
@Alyosha As I understand residues in $\Bbb C \cup \{\infty\}$, residue of $f(z)$ at $z = \infty$ is the residue of $-1/z^2f(1/z)$ at $z = 0$. This is explained if you think of the contour-integral definition of residues. [$f(1/z)d(1/z) = -1/z^2f(1/z)dz$]
'doing math' --- my girlfriend calls it 'playing math', which has always sort of bothered me... if it's the weekend and i'm working on math, she rolls her eyes and says, 'oh, you're playing math, call me next month', and i'm like ok awesome, meanie.
'playing serious math'
alright, here's a bad math joke and then i actually do need to go do some math :
@Danny I don't know but maybe you can show that the series converges and then that the derivative also converges and that the derivative equals the original sum. Then maybe apply the theorem that gives uniqueness of solution of a differential equation.
@Hawk Show that if the formula holds for two polygons that have a common side, it also holds for the union of these polygons. That allows you to reduce the general case to the case of triangles.
@Hawk as ronno says, this looks like Pick's Theorem, but in your question, you never say that $Z$ is the number of interior lattice points and $B$ is the number of boundary lattice points.
@BalarkaSen Did you see this one? $$\sum_{n=1}^{\infty} (-1)^{n+1} \left(\log(2)-\frac{1}{n+2}-\frac{1}{n+3}-\cdots -\frac{1}{2(n+1)}\right)=\frac{1}{8} (2 \log(2)-4+\pi)$$
