@Calculus It is the length, that means that if $\gamma : [a,b] \rightarrow \mathbb{C}$ differentiable, we have that $L(\gamma) = \int_a^b |\gamma'(t)| dt $
hope you don't mind me bothering you guys - I've got a probably really stupid misconception here - my binomial expansion for (1 + 2x)^-1 to three terms is different to the binomial expansion of (1 + 2x)^-2 with a coefficient of (1 + 2x) added after expansion (i.e. [1+2x]/(1+2x^2) with the denominator expanded) to three terms - why is this? :( I thought (1 + 2x)/(1 + 2x)^2 was equal to (1 + 2x)^-1
While browsing through 1-answered questions, came across this question which had a wrong answer (got deleted when I pointed it out), and I tried answering.
Can someone please check the answer? I came upto convergence and pointed to Alpha. I feel it's incomplete. Can someone help?
Think of it like this @Calculus you will find the least value that will show that the unit circle will be engulfed in it (subset) and hence you will be able show... I think
Sorry I didn't notice your message - I went from $ \frac{(1 + 2x)}{(1 + 2x)^2} $ to $ \frac{1}{1 + 2x} then found the three terms by 1 + (-1)(2x) + (-1)(-2)/2.1 * 4x^2 to get 1 - 2x + 4x^2
I wish to have a function that maps [0,1] to [0,1]. I also require that f(0) = 0 and f(1) = 1.
Also, I would like the function to be of sigmoidal shape, such as this:
The above function is ok, but I would like the function to be flatter at the bottom and the top, and have a steeper ascent....
I have seen it before, and I don't think it's of much use.
I mean, if that proof could be generalized to the fundamental theorem for general UFDs (integral closures of algebraic number fields, say), I'd have cared. No point in going high-brow about number theory over $\Bbb Z$ (which is easy, in general).
If you can prove the fundamental theorem of arithmetic over Z using Jordan-Holder for groups, I'm sure you can use Jordan-Holder for modules math.hawaii.edu/~lee/algebra/jordan.pdf on UFD' I'm sure?
$R-\mathsf{Mod}$ is a much simpler category than $\mathsf{Ring}$. It's not apparent to me how you could apply things about modules to prove things about rings.
In any case, there are lots of non-UFDs out there, so I believe anything that's true in general for R-modules fails to say something interesting for rings. Of course, this is what I believe, and I'd be happy to know I am wrong.
(to emphasize what I'm saying : subobject structure of modules is much easier than subobject structure of rings. consider Noetherian modules and Noetherian rings, say)
@BalarkaSen Sure, I just mean the fact that it generalizes from groups to modules probably means the same idea can be used for UFD's, as that paper says
@Bib what does universal with respect to this property mean in this context: "In an additive category $\mathcal{A}$ a kernel of a morphism $f: B\to C$ is defined to be a map $i : A \to B$ such that $fi = 0$ and that is universal with respect to this property."
@EnjoysMath You have proved that $Z_n(\bigoplus_i A_i) = \bigoplus_i Z_n(A_i)$ and $B_n(\bigoplus_i A_i) = \bigoplus_i B_n(A_i)$. $B_n$ is normal in $Z_n$, so $\bigoplus Z_n(A_i)/\bigoplus B_n(A_i) \cong \bigoplus (Z_n(A_i)/B_n(A_i)) = \bigoplus H_n(A_i)$.
@Remember $U, V$ be open subsets of a top. space $X$. $f : U \cup V \to Y$ be a map between top. spaces. prove that if $f$ restricted to $U$ and $V$ are both continuous, then $f$ is continuous.
I did all exercises in Simmons,till topolgical spaces.... Finished topological spaces in munkres ... Now doing product spaces which is just after topological spaces
Balarka is just trying to make sure you know the ins and outs, the product and box topologies are different in the infinite case. That's not important now. The gluing lemma lets you join two functions together given they agree where they overlap. It's not needed now either.
I have a matrix integration problem. It is based on the first integral under the section, "energy transfer efficiency and transport time" in the article, environment-assisted transport. There is a function, $\rho(t)$ that is the most important in calculating the integral but is a time-dependent m...
Let $u(x,y), x^2+y^2 \leq 1$, a solution of
$$u_{xx}(x,y)+2u_{yy}(x,y)+e^{u(x,y)}=0, x^2+y^2\leq 1$$
Show that $\min_{x^2+y^2 \leq 1} u(x,y)= \min_{x^2+y^2=1} u(x,y) $.
We suppose that $\min_{x^2+y^2 \leq 1} u(x,y) \neq \min_{x^2+y^2=1} u(x,y) $.
At the solution it is said that since $\{ (x...
@DanielFischer Hi, reading my exam on complexe analysis (what we talked about last time) it was to write the holomorphic function $f$ for the bounded domain $\Omega$ to $\Omega$ as $$f(z)=z+c(z-a)^k+(z-a)^{k+1}h(z).$$ I wrote that $h$ is holomorphic on a neighbourhood of $a$, but it was asked to prove that $h$ is, in fact, holomorphic on $\Omega$
Many online journals, or journal repositories, are accessible if you have institutional access. Being e.g. on college wifi gets you such access. That's all.
@Gato Well, $$h(z) = \frac{f(z) - z - c(z-a)^k}{(z-a)^{k+1}}$$ is evidently holomorphic on $\Omega\setminus \{a\}$. The previously mentioned things show it is holomorphic at $a$.
@DanielFischer Ok, I did this but It was asked in another question, so perhaps I missed something else. Another question : to compute the maximum of $\sin(z)/z$ on the unit circle, I used fist the maximum principle to say that it's at $\vert z \vert =1$ but how can I continue ? I now that $\sin(z)=z-z^3/3+\cdots+ o(z^{2n+3})$
@Gato We need an absolute value somewhere, for a complex-valued function, it doesn't make sense to talk of a maximum, so $\max \left\lvert \frac{\sin z}{z}\right\rvert$ is attained at $i$ and $-i$.
@DanielFischer Right, badly expressed. So to write correctly a solution is : By the maximum principle $$\max_{z\in D}|f(z)|=\max_{|z|=1}\frac{|\sin z|}{|z|}=\max_{|z|=1}|\sin z|.$$
Furthermore $\lvert \sin z\rvert = \left\lvert\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}z^{2n+1}\right\rvert \leqslant \sum_{n=0}^\infty \frac{\lvert z\rvert^{2n+1}}{(2n+1)!} = \sinh \lvert z\rvert$ by the triangle inequality, then I have equality for purely imaginary z, so the maximum of $\vert\frac{\sin(z)}{z}\vert$ is attained at $i$ and $−i$.
Is it sufficient to get all the points in an exam ?
@DanielFischer Ok, thanks. I am working with the following exercise : Let $f$ be an entire function, such that $\vert Im(f(z))\vert -\vert Re(f(z))\vert <0$. I need to prove that $f$ is constant on $\Bbb{C}$, pretty sure I need to use Liouville's theorem. I plot the graph of the inequality but not sure how I can use the hypothesis.
@Gato You need to transform it a bit before applying Liouville's theorem. If $f$ is entire and there is a $w\in \mathbb{C}$ and an $\varepsilon > 0$ with $\lvert f(z) - w\rvert \geqslant \varepsilon$ for all $z$, what can you do to make Liouville's theorem applicable?
I wish to have a function that maps [0,1] to [0,1]. I also require that f(0) = 0 and f(1) = 1.
Also, I would like the function to be of sigmoidal shape, such as this:
The above function is ok, but I would like the function to be flatter at the bottom and the top, and have a steeper ascent....
Let $u(x,y), x^2+y^2 \leq 1$, a solution of
$$u_{xx}(x,y)+2u_{yy}(x,y)+e^{u(x,y)}=0, x^2+y^2\leq 1$$
Show that $\min_{x^2+y^2 \leq 1} u(x,y)= \min_{x^2+y^2=1} u(x,y) $.
We suppose that $\min_{x^2+y^2 \leq 1} u(x,y) \neq \min_{x^2+y^2=1} u(x,y) $.
At the solution it is said that since $\{ (x...
