Hi. I'm looking at Milnor's "Geometric Realization" paper. I'm confused as to how he's defining the homology of a simplicial set. One way is to define it to be the homology of the realization, the other is to take the free abelian group construction and then take the homology of the alternating chain complex associated to this. It's not clear to me what he means?
He writes $H_(K) \cong H_*(S \vert K \vert)$, but does he mean $H_*( \vert K \vert) \cong H_ (\vert S (\vert K \vert) \vert)$
Let $A$ be the reals from [0,100] inclusive. I want to find the maximum and minimum sum of $x,y,z \in A$ s.t $xyz=64$. I am using lagrange multipliers and since I have a closed and bounded set it should get me both a min and max; however, I am only getting a min. How do I get the max using lagrange multipliers?
My work so far:
Let $f(x,y,z) = x + y + z$ and $g(x) = xyz - 64$ so $ \nabla f = [1,1,1] $ and $ \nabla g = [yz, xz, xy] $. $\nabla f = \lambda \nabla g$
So we get that $\lambda = \frac{1}{yz} = \frac{1}{xz} = \frac{1}{xy}$ and thus $x=y=z$ so $x=y=z=4$
This is what minimizes $f$, however, how do I find what maximizes $f$?
I can see easily from inspection that the values that will maximize $f$ is $x= 64, y=1,z=1$. However, I am not understanding how to get these values from lagrange multipliers
Hello there! I my script the unparameterized curve is given by $tan(x)=(2/3)*sqrt(abs(t)^3)*sgn(t)$. They rearrange the equation to get: $t= sgn(x)*abs((3/2)tan(x))^(2/3)$. I am trying to figure out how to obtain the second equation and get stuck at the following step: $sgn(x)*abs((3/2)tan(x))=t*sqrt(abs(t))$.
I have no idea how to proceed (as the domain must include negative “t”)... please help!
let $f_1(x)=f(x)$ $f_2(x)=f(f(x))$ $f_3(x)=f(f(f(x)))$ and so on.... is there some function for which $f_{\infty}$ converges? except for something trivial like $f(x)=1^x$
Well, that example I gave you technically converges on the whole real line, because $\lim\limits_{n\rightarrow\infty}\frac{x}{2^n}=0$ for all $x\in\mathbb{R}$
Pointwise convergence, I believe is the term
There is also uniform convergence, which is stronger than pointwise
(Also, sorry if my previous message was unclear. Reading it back it seems like I'm implying that my example isn't a definable function)
Well, first compare the values of $f_{2n}(x)$ and $f_n(f_n(x))$
This is how you are defining this sequence, and so, if it doesn't hold for any particular $n$, then these $f_n$ don't qualify as one of these sequences.
I have question about equality in Cauchy-Schwarz inequality:
I have this proof with me:
In proof, they only show that equality holds if $x$ is a specific scalar multiple of $y$, namely $\frac{||x||}{||y||}$, but in the statement of theorem, they claim that equality iff one of them scalar multiple of the other.
let $f_1(x)=f(x)$
$f_2(x)=f(f(x))$
$f_3(x)=f(f(f(x)))$ and so on...
Is there some function $f(x)$ for which $f_{\infty}(x)$ is non-constant and converges? It is okay if the $f_{\infty}(x)$ you find has a finite radius of convergence.
The only example that comes to my mind would be like,...
I'm trying to calculate the limit of $\frac{x^4 y^2}{x^4 + y^2}$ as $(x,y) \to (0,0)$. Is it valid to just say $0 < \frac{x^4 y^2}{x^4 + y^2} \leq \frac{x^4 y^2}{x^4} = y^2$ and take $y \to 0$?
although this makes me question that if the antiderivative of the function cannot be expressed in the form of elementary functions then does that imply that the second antiderivative cannot be expressed in the form of elementary functions?
let $f_1(x)=f(x)$
$f_2(x)=f(f(x))$
$f_3(x)=f(f(f(x)))$ and so on...
Is there some function $f(x)$ for which $f_{\infty}(x)$ is a continuous non-constant function that converges? It is okay if the $f_{\infty}(x)$ you find has a finite radius of convergence.
The only example that comes to...
I think you mean to say that the infinite function composition converges on a particular function, not that $f_{\infty}(x)$ converges. Is that accurate?
The identification of the two punctured disks $\iota_{1,2}$ and the retraction from the collared neighbourhood onto the boundary of the disk make the identification of the boundaries so it's all continuous
$C_n(A+B) \to C_n(A\cup B)$ is a quasi-isomorphism. Is the natural map $\cfrac{C_n(X)}{C_n(A+B)} \to \cfrac{C_n(X)}{C_n(A\cup B)}$ also a quasi-isomorphism?
Where $A, B$ are open subsets of $X$ and $C_n(A+B)$ denotes small chains (consisting of simplices whose images lie either entirely in $A$ or entirely in $B$).
@MikeMiller So, I look at the long exact sequence in homology induced by the short exact sequences: $0 \to C_n(A+B) \to C_n(X) \to C_n(X)/C_n(A+B) \to 0$ and $0 \to C_n(A\cup B) \to C_n(X) \to C_n(X, A\cup B) \to 0$.
Yes but you should probably figure out precisely how the five lemma applies. You need to manipulate these a little to break it into short exact sequences.
Quick sanity check: $\Bbb C\setminus \lbrace 0 \rbrace$ is not a star domain wrt any point in $\Bbb C\setminus \lbrace 0 \rbrace$ because $0$ always lies on the line connecting $z$ and $\bar{z} - 2\operatorname{Re}(z)$... right?
It's sort of like one of those group isomorphism theorems. Phrased much more generally, suppose you have chain complexes $C_1, C_2, C_3$, and inclusions $C_1 \hookrightarrow C_2 \hookrightarrow C_3$. Suppose the first inclusion is a quasi-isomorphism.
Then the same argument implies that $C_3/C_1 \to C_3/C_2$ is a quasi-isomorphism.
Sounds like a simple version of the third isomorphism theorem to me.
yeah it's exactly the third isomorphism theorem + long exact homology sequence. if $C_1 \hookrightarrow C_2$ is a quasi-isomorphism, then $C_2/C_1$ is acyclic. So if you apply the LES to $0 \to C_2/C_1 \to C_3/C_1 \to C_3/C_2 \to 0$, you get what you want
I'm just trying to figure out this word for my own notes. I'm doing a presentation as part of my Economics Masters. I'm going to draw the function inside $\frac{\mathrm d}{\mathrm dx}\left(1-\exp\left(-\int_0^x \theta(s)\,\mathrm ds\right)\right)$ on the board, but I was wondering to myself what to call it.
> if $C_1 \hookrightarrow C_2$ is a quasi-isomorphism, then $C_2/C_1$ is acyclic.
@MatheinBoulomenos So, this follows by considering the LES of $0 \to C_1 \to C_2 \to C_2/C_1 \to 0$, which is: $\cdots H_n(C_1) \to H_n(C_2) \to H_n(C_2/C_1) \to H_{n-1}(C_1) \to H_{n-1}(C_2) \to \cdots$.
you don't even need the 5-lemma with this approach. Just apply LES to $0 \to C_1 \to C_2 \to C_2/C_1 \to 0$, then apply LES to $0 \to C_1/C_2 \to C_3/C_1 \to C_3/C_2 \to 0$ and then you're done
I am trying to transform $y' - \frac{1}{x}y = y^2 - \frac{3}{x^2}$ into a second order linear ODE. According to math.stackexchange.com/a/3053787/109355 I should use $u = -\frac{y'}{y}$, but I'm having trouble algebraically eliminating every $y$ with this substitution. How do I get the ODE in terms of $u$?
Say one has a polynomial $x^4+x^3y^1+x^2y^2+...+y^4$ which is a sum of $x_i y_j$ over all indices $i,j$. Can the Cauchy Schwarz inequality be applied to that sum, considering the fact that the sum is being taken over two different indices?
The whole point is somehow that when you integrate something which has a convergent power series expansion with possibly negative power terms (could be infinitely many negative power terms, I am allowing essential singularities), only the coefficient of the $1/z$ term is picked up
That's exactly the working principle of Cauchy integral formula
Exactly. Somehow amongst all $z^n$, $n \in \Bbb Z$, only $n = -1$ is the unusual guy in the sense that all the others have antiderivatives which are honest to god functions. But the $1/z$ guy integrates to $\log(z)$ - something multivalued.
Which has value well-defined only upto multiples of $2\pi i$ - how many multiples are there is recorded by the winding number of the loop you are integrating around
OK, my non-exactness thing wasn't the right condition. I meant I want a form which is nonexact and it's nonexactness is witnessed by the equator $|z| = 1$, as in, it integrates to something nonzero along the equator.
Basically the divisor corresponding to it should be $a(0) + b(\infty)$ with $a + b = -2$ and $a, b < 0$
Basically non-exact on both the charts containing $0$ and $\infty$
@TedShifrin it can mean lots of things, here I guess its "lecture notes"
that would be students slang
(whatever syllabus means)
Anyone interested to share a thought on the Riemann-Hypotheses? I just came to the surprising idea that no one probably really understands it. Because if so we would have a proof, right? Does that conclusion always work? What about the four-color theorem then.
Just some thought after the first evening beer ...
yes, depending on how something is proved, we might have little understanding on "why" something is true even if we have a proof
if you proof something by brute force/random ad-hoc tricks that give no insight, then I guess we don't really understand the result, even if we know it's true
I agree with what @Mathein said. In terms of teaching, in more elementary courses I only presented a proof when it gave understanding to how/why the theorem works.
@Rudi_Birnbaum certainly some things are known. pretty sure there's entire papers written under the heading of "Suppose the Riemann hypothesis is false."
@Rudi_Birnbaum yes, we have a pretty good understanding on the implications of RH and the failure of RH I'd say. The zeroes of the zeta function are linked to the distribution of primes
There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false?
But understanding would not mean to see why one counterexample works it would rather give rise to a massive series of conclusions, or something like that.
@TedShifrin yeah maybe then not really all but lets say at least almost all ...
Maybe we can give an example where a proof really lead to a breakthrough in insight, that has not existed when the thing was a conjecture?
In modern maths the situation seems to be a bit twisted, such that you first need the breakthrough in insight/methods that is then followed by the proof
But I guess that situation might be seen as "in principle equivalent"
@Semiclassical Well that looks also cool "Thus if RH is false, any positive integer can be the class number of finitely many imaginary quadratic fields."
Before the work of Heilbronn, Mordell had shown that if infinitely many imaginary quadratic fields have the same class number (any common value) then RH for the Riemann zeta-function is true. Thus if RH is false, any positive integer can be the class number of finitely many imaginary quadratic fields. — KConradJul 11 '13 at 22:33
If you read the wiki biography of Riemann its kind of weird how he died so young and didnt do mathematics for all that long, but still so many modern ideas and concepts "go back to something Riemann was thinking about"
also his famous habilitation talk supposedly contains the germ of so much stuff. That doesnt really seem all that believable to me, how long did he talk for and how vague must the talk have been for those claims to be true
Suppose I have a subring R of $\mathbb{C}[x]$ with the property that $f(1) = f(-1)$ for all $f \in R$. Can anyone show me or give me a hint to show that R is not principal ideal domain?
the mathematical anecdote re: habilitation which I keep looking for (and keep failing to find)
is where one of the members of the habilitation committee asked the speaker, of the three topics they were offering to speak about, which they knew the least about
and subsequently asked them to speak on that topic, to see what they'd say about a topic of which the speaker knew nothing
I looked at some example tree once. Some guy had among his academic ancestors: Gauß, Euler, Hilbert, Fourier, Kant, Fichte, Schelling, Hegel and several other well-known names
let $f_1(x)=f(x)$
$f_2(x)=f(f(x))$
$f_3(x)=f(f(f(x)))$ and so on...
Is there some function $f(x)$ for which $f_{\infty}(x)$ is non-constant and converges? It is okay if the $f_{\infty}(x)$ you find has a finite radius of convergence.
The only example that comes to my mind would be like,...
There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false?
