1:44 AM
Hi @loch, @Ted

hi a @Balarka
Is it actually morning there?

Yeah 7:12 AM lol

Must be 7:15 ... the half hour always throws me off.

Regarding your comment here, $[Y]$ would in general be represented by a continuous image of a delta complex, and it seems believable that you can write any delta complex as quotient space of a single simplex.

Hi @BalarkaSen @TedShifrin

1:49 AM
Hmmm, how do I give the fundamental class of a many-holed torus with one?
hi @loch
Or maybe I need something higher-dimensional.

@TedShifrin The surface of $g$ is quotient space of some fundamental polygon, so no problem.
Map a simplex homeomorphically to the polygon.

This doesn't sound believable in general.
how do you even do $\Bbb CP^n$?

That's quotient of a sphere, which is quotient of a disk - map a simplex homeomorphically to a disk!
You can even do that simplicially.
I'd like to see an example of a delta complex $D$ such that there is no simplicial quotient map from a triangulation of $\Delta^n$ to $D$. I think this is always true.

OK, we need something with a lot more topology.
I've never heard of such a thing.

Me neither but sounds like a cool question

1:54 AM
Meanwhile, the OP was confuzled by my comment.
Perhaps you can aid him.

Intuitively it feels that you can break $D$ apart into simplices and rearrange them appropriately to get a triangulated $\Delta^n$ with lesser identifications.

@TedShifrin are you planning to teach any more AoPS courses in the near future?

@TedShifrin I can write an answer but then I have to sneak it under the rug that complex projective varieties admit triangulations ;)
I don't actually know a proof of that. If one can get a Whitney stratification, that's enough, by a theorem of Goresky. I don't know how to Whitney stratify a general complex projective variety: the singular stratification need not be one (eg the Whitney umbrella). I think one has to pass to some appropriate finer stratification than the singular one, but I don't know the details at all

2:34 AM
@Daminark Thank you very much.

2:49 AM
Good morning everyone! I had a small doubt in the application of Leibniz integral rule
How do I differentiate $\int_0^x tf(x-t)dt$ where f(x) is a function.
With respect to x

1 hour later…
3:53 AM
@TedShifrin Anyway, so topologically, any $n$-dimensional connected finite CW complex $X$ can easily be seen to be a ball quotient. Impose a CW structure with a single $0$-cell: this is always possible. Then $\varphi_i : D^n \to X$ be the characteristic maps of the various $n$-cells; $\bigvee_i \varphi_i : D^n \vee \cdots \vee D^n \to X$ is all of them at once, sending the wedge point to that single $0$-cell. This is a quotient map. Now a wedge of a bunch of $D^n$'s is a quotient of $D^n$.
I don't know how to do this simplicially though, because if I have a quotient map $D^n \to X$, I can $\epsilon$-homotope it to make it simplicial modulo appropriate simplicial structures on domain/codomain, but this need not be a quotient map any longer.

Let $G$ be a group with identity element $e$. And $N$ be a normal subgroup. Let $[G:N]=12$. Then $x^{24}\in N$ for all $x\in G$.
I think this is because, for any $x\in G$, $x^{12}N=(xN)^{12}=N$, hence $x^{12}\in N$ hence $x^{24}\in N$. Right?

(Well, there's a small problem above because the image of the attatching maps might not be all of the $(n-1)$-skeleton but that's easily fixable: you can homotope the attatching maps so that it goes "round and back again" around it)

1 hour later…
5:23 AM
Ok, so this problem has a nice solution, actually. I have a simplicial complex $K$ and I want a triangulation on the $n$-simplex such that $f : \Delta^n \to K$ sends $i$-simplices to $i$-simplices. Basically this is going to be a sort of "unfolding" of $K$ (think of unfolding a cube into a cross of squares).
Just take the "dual graph" to $K$, which is given by joining the centers of all the $n$-simplices by edges. Take a maximal tree of this, and mark the simplices of $K$ which doesn't intersect this maximal tree. Cut $K$ along those simplices to get a simplicial complex $K'$ from which $K$ is obtained by gluing some pairs of faces togather. $K'$ deformation retracts to the maximal tree, so is topologically a ball.
Put up a homeomorphism $\Delta^n \to K'$ and pull the triangulation back. You have obtained $K$ as a simplicial quotient of $\Delta^n$.

duality yay

5:34 AM
By simplices above I meant the (n-1)-simplices, i.e., the faces. The idea is you cut along faces to get something which is a ball

6:24 AM
I am trying to show that if $K$ is a simplicial complex and the geometric carrier of $K$ has $k$ many path components, then the zeroth homology group of $K$, i.e., $H_0(K)$ is isomorphic to $\mathbb{Z}^k$.
My only observation so far is that the elements of $C_0(K)$ (the $0$-dimensional chain group of $K$) is isomorphic to $\mathbb{Z}^{|\text{vert}(K)|}$ where $\text{vert}(K)$ denote the set of all vertices of all simplexes in $K$.
However, I don't know how can I related this to $k$ (the number of path components of $K$). Any ideas?

When are two (singular) 0-simplices the border of a (singular) 1-simplex?

@AlessandroCodenotti: Is this comment in response to my comment(s)?
(Sorry if it is not. But I just thought that I should confirm.)

6:39 AM
@AlessandroCodenotti What are singular 0-simplices? I don't know about it.
However I think that $0$-simplexes are "endpoints" of $1$-simplex iff each of the simplex is a vertex of the one simplex.
I am not really fond of geometry, so I was trying to think of a purely algebraic solution of the question is possible.

How to generate standard normal coefficients in excel?
Basically I am doing Functional Data Analysis. Where I need to calculate a polynomial which is a function of time and has coefficients which are from standard normal distribution.
should I just generate a random number between 0 and 1 in excel for it to be from standard normal distribution?

I guess this answers the question
In the question I uploaded I have one query
How do I calculate the variance
in order to calculate the error term
I know what is beta of t what is the beta of s in the equation defining variance
Also what is E[X(t)X(s)] ?
is E the expected/average value of X(t) and X(s) ?

7:05 AM
@user170039 how do you define $H_0$?
(or $H_n$ in general)

@AlessandroCodenotti Same as here.

Which chain complex are you using?

Do you mean the definition of $C_n(K)$ for all $n\in \mathbb{N}$?

Yes
(well to be precise the chain complex is the whole of the $C_n$ and maps between them)

Let $K$ be a oriented simplicial complex. Denote by $\bf{O}_n(K)$ the set of all oriented $n$-simplexes of $K$. Let $f:\bf{O}_n(K)\to \mathbb{Z}$ be a map such that $f(+\sigma)+f(-\sigma)=0$ for all $n$-simplex $\sigma\in K$.
Call such a map a $n$-chain. For $n\in \mathbb{N}$ define $C_n(K):=\{f:f~\text{is a}~n\text{-chain map}\}$.
Let $+:C_n(K)\times C_n(K)\to C_n(K)$ be defined in the usual way of function addition.
Do I need to elaborate the boundary homomorphisms also?

7:29 AM
Well what is the boundary of a $1$-chain? To compute $H_0$ you need $\ker\partial_0$, but that is the whole of $C_0(K)$, so you only need to figure out what's the image of $\partial_1$

7:47 AM
As per the notes which I am reading $\partial_q(\widehat{+\sigma^q})=\displaystyle\sum[\sigma^q,\sigma^{q-1}]\widehat{+\sigma^{q-1}}$ where $\widehat{+\sigma^{q}}$ is the elementary $q$-chain (defined below) and $[\sigma^{q},\sigma^{q-1}]$ is the incidence number.
The elementary $p$-chain is defined as follows, $$\widehat{+\sigma}(\tau)=\begin{cases}1&\text{if}~\tau=+\sigma\\-1&\text{if}~\tau=-\sigma\\0&\text{else}\end{cases}$$
Here the sum runs over all the positively oriented $(q-1)$-faces of $\sigma^q$.
This definition is then extended by linearity to whole $C_q(K)$

2 hours later…
9:29 AM

### JEE Maths Zone

3 hours later…
12:34 PM
The following expression reduced is apparently equal to x, can anyone explain how that is the case?
$$\frac{x^2+2x}{x+2}$$
When I reduce the expression, I get the answer to be equal to $x^2$. This is how I would go about reducing it:
What did I do wrong?

@SebastianAlexanderBNielsen you can not cancel the 2s like that.

Hi, I have a small question:

Does $\neg (f (a,b) \neq x )\equiv (f(a,b) = x) \equiv(\neg f(a,b)\neq x)$ hold, where $f$ is a mapping and $a,b,x$ are elements of the underlying universe?

Step 1 factor the x out of the numerator
Step 2 divide the numerator and denominator by (x + 2)

But how? I can "break apart" the denumerator $x+2$.
uh wait, thanks

12:50 PM
np

1:01 PM
Hello, this seems to be a bit problematic for me

https://math.stackexchange.com/questions/3231729/if-sum-a-n-converges-then-sum-a-n-fracnn1-converges-as-well?noredirect=1#comment6648204_3231729
I don't know whether or not it will be valid to write $\lim_{n \to \infty} \frac{a_n^{n/(n+1)}{a_n}$
$\lim_{n \to \infty} \frac{a_n^{n/(n+1)}{a_n}$
$\lim_{n \to \infty}$ $\frac{a_n^{n/(n+1)}}{a_n}$
=1

1 hour later…
2:30 PM
Does someone know how to refer to a system that consists of several equations in IEEE? For example i have a system described by equation (1) and a controller for system (1) described by equation (2). I basicaly want to refer to the whole dynamics (1) and (2). I was thinking if i could say: Dynamics (1,2).

1 hour later…
3:52 PM
May 11 at 16:52, by Tobias Kildetoft
If you lecture long into the void, the void starts lecturing back.
[Random]
Problem solving surface: The Path of sudden least resistance

Sometimes given a mathematical problem. Suppose we can represent it as a n dimensional surface, and let some point p be the solved state of the problem

In the question I uploaded I have one query
How do I calculate the variance
in order to calculate the error term
I know what is beta of t what is the beta of s in the equation defining variance
Also what is E[X(t)X(s)] ?
is E the expected/average value of X(t) and X(s) ?
If anyone can even provide the approach to solve that question it would be helpful

Let p be a set of curves that can be plotted on the surface. For some problem which is very difficult, there exists highly non smooth surfaces such that only certain curves will lead to p
This thus translate to a highly specific idea or understanding, such that the hard problem becomes straightforward to solve
What may be interesting is that: Given a problem surface, when will solutions to them be solution sets of finite cardinality or even of cardinality 1
@noob I do not know the other question, but E[X(t)X(s)] is going to be the expectation value of the product X(t)X(s)

3 hours later…
6:37 PM
Is "modular arithmetic", i.e. Peano arithmetic where $\neg \exists x (Sx = 0)$ is replaced with $\exists (Sx = 0)$ $\omega$-inconsistent?

7:19 PM
Why is the falling factorial $(m)_n=\frac{m!}{(m-n)!}$? I'd really appreciate a derivation. I define the falling factorial as given on Wikipedia.

$m!$ is the product of all integers from $1$ to $m$. $(m-n)!$ is the product of all integers from $1$ to $m-n$. So what's left after diving $m!$ by $(m-n)!$ is the product of all integers from $m-n+1$ to $m$.

7:38 PM
@Thorgott Nice. Thanks!

8:33 PM
What does it mean for a function to be defined on an open neighbourhood of a point? I'm familiar with the term without the word 'open' or 'open' being replaced by 'punctured'. Does it simply imply an open interval around that point?

@schn in a general topological space you have neighborhoods

GOD it feels nice when you draw a commutative diagram
3

@schn a subset $A$ of $\Bbb R$ is open iff every point $x \in A$ is contained in an open interval $(x-\varepsilon,x+\varepsilon)$ which in turn is contained in $A$
so you have a "full set of points" near each point
@BalarkaSen ikr

@LeakyNun Alright, so saying a function is defined in an open neighbourhood at a point would mean an open interval $(a,b)$ which contains that point? Thanks!

8:48 PM
@BalarkaSen D:

lmao
i mean it's more like the joy you feel when you're crushing a cockroach under your shoes

I have to draw a commutative diagram and take a direct limit in my set theory seminar talk in a couple of days, I'll bring shame upon my family and my cows

@schn the former would imply the latter and indeed the latter would imply the former...

9:23 PM
If a real valued function $C^1$ (or maybe just differentiable is enough) has a derivative bounded by $L$, does that mean that $L$ is a Lipschitz constant for the function? Or is it possible you need a larger number than $L$?

@GFauxPas $L$ is enough. Differentiable is enough.

Thanks, useful to know

9:50 PM
I can use help with this question
The system is $\begin{bmatrix} \dot x \\ \dot y \end{bmatrix} = \begin{bmatrix} t^{-1} & 0 \\ 1 & 3t^{-1} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$
I have the general form of the solution
$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} c_1 t \\ c_2 t^3 - c_1 t^2 \end{bmatrix}$
I have a matrix $\Pi(t,1)$ with two columns spanning the solution space satisfying $\Pi(1,1) = I$
$\Pi(t,1) = \begin{bmatrix} t & 0 \\ t^3 - t^2 & 1 \end{bmatrix}$
now the problem is:
Solve $\begin{bmatrix} \dot x \\ \dot y \end{bmatrix} = \begin{bmatrix} t^{-1} & 0 \\ 1 & 3t^{-1} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \end{bmatrix}$
since that's part 2 of the problem, I assume that $\Pi(t,1)$ is supposed to play a role here
I don't think I'm supposed to set up a system of two equations, I think it's supposed to be simpler than that

@BalarkaSen I thought this was illegal in India

oh I found the theorem
interesting, the inhomogeneous part of the solution is given by
$\displaystyle \int_{t_0}^t \Pi(t,s)g(s) \, \mathrm ds$ where $g(s)$ is the the inhomogeneous part, in this case $(1,0)$
that's interesting

@ryan cows, not roaches

10:11 PM
@BalarkaSen there was a guy in the hbar who denouncing killing of bugs
I believe he had authority to speak for the whole country

that does sound very indian to me
everyone here has authority to speak for the whole country
@RyanUnger Why are you lurking in "Jee Advanced physics solving strategies"

If you want to understand a complex function do you study it's roots and poles?

10:28 PM
In this answer math.stackexchange.com/questions/3230803/finding-borel-sets by Daniel Wainfleet, why $f^{-1}(U\cap \Bbb R)=g^{-1}(U\cap \Bbb R)$

@Ultradark that's sufficient for something like $f(z)/g(z)$ where $f(z),g(z)$ are polynomials in $z$. that's a very common scenario
however, that will not help you understand something like $f(z)=\ln z$ very well. The behavior at zero is neither that of a root or a pole, but of a branch point
$f(z)=e^{1/z}$ is another example where things go weird at zero in a way other than a pole or a root

@Semiclassical Is it understood this weirdness?

I think it's not true because U might contain the point 0, thus $g(x)\not\in U$, am I correct?

There's a theorem that, correct me if I get the hypotheses incorrect
that if $f(z)$ is holomorphic except at a point $z_0$ where it has an essential singularity, than $f(z)$ attains every complex number in a neighborhood of that point, with possibly one exception

@Ultradark the classification is understood: en.wikipedia.org/wiki/…
(note that a root is a place where $f(z)$ is well-behaved, so it's not a singularity)

10:36 PM
that's the case with $e^{1/z}$

@GFauxPas yeah, picard's theorem (i forget if it's the little or big one)

Great Picard's Theorem: If an analytic function f has an essential singularity at a point w, then on any punctured neighborhood of w, f(z) takes on all possible complex values, with at most a single exception, infinitely often.
which is pretty damn cool

that means all zeros of the analytic function are also mixed in

another fun example: The series $f(z)=z+z^2+z^4+\cdots+z^{2^n}+\cdots$ converges everywhere inside the unit circle. But it has no analytic continuation outside the unit circle!

That's a typo there Semi
$z^+$

10:40 PM
thx

that's cool. What about on the circle?

doesn't converge anywhere on the unit circle iirc. the unit circle thereby serves as a "natural boundary of analytic continuation".

@GFauxPas convergent series tend to 0

16

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem intimidated me when I first saw it as an undergrad, as the series under the root isn't the expansion of a...

$z^{2^n}$ doesn't tend to 0

10:41 PM
does the essential singularity encode information about the analytic function or something?

oh right
durr
if it has an essential singularity it's not analytic there

@LeakyNun not on the unit circle, anyways

but it will be expressible as a laurent series on an appropriate annulus

$$\sum_{n=1}^\infty \frac1{2^n} = \sum_{n=1}^\infty \frac1{2^n \ln(2^n)}$$

10:45 PM
@LeakyNun how is that an equivalence?

@Semiclassical can we find the pdf of Y=g(X) where X~Uniform(-1,1) and... g(x)=sin(1/x)?

0

I was searching for primes of the form $n^{n^2}-k$ on PARI/GP and noticed that primes of this form for same $k$ are quite rare. The probability of finding a prime of this form is $\frac {1}{n^2 \log (n)}$, which makes me believe that the number of primes for same $k$ of this form would be finite....

any ideas?
guys

if X has a pdf, what are sufficient conditions on g such that g(X) also has a pdf?

what is a punctured neighborhood of $w$

@LeakyNun "$g$ is a diffeomorphism" :)

10:59 PM
Neighborhood with a hole in the middle
One point removed in the interior

You can obtain a valid pdf of $g(X)$ whenever $g$ satisfies the necessary conditions on the range of $X$ for the change of variables formula to hold.
Eg, $g^{-1}$ exists and is injective on $\text{Range}(X)$, is $C^1$ and $dg^{-1}$ is full rank.
You can weaken these assumptions, of course, because a pdf is well-defined upto a measure zero set.
So for example if you start out with $X \sim N_2(\mathbf{0}, I)$ standard multivariable normal, and $g(x, y) = (\sqrt{x^2 + y^2}, \tan^{-1}(y/x))$ which is defined on the whole plane minus the non-negative real axis, say, then $g(X)$ is still absolutely continuous. Simply because even though $g$ wasn't defined on the full range of $X$, it was defined on a full measure subset, and $g^{-1}$ is a local diffeomorphism (it's a covering map we all know and adore)
We picked out a large local section of the covering map defined on a full measure open set, that's all. Incidentally, $g(X)$ turns out to be jointly distributed as $(R, \Theta)$ where $R$ and $\Theta$ are independent (!) and $\Theta \sim \text{Unif}(0, 2\pi)$ (!), indicating the rotational symmetry of the standard bivariate normal. $R$ is the "radial part" which is called the Rayleigh distribution; it's squareroot of $\text{Exp}(1)$.
Speaking of square roots, a way simpler example: $g : \Bbb R \to \Bbb R$, $g(x) = x^2$. $g^{-1}$ is piecewise defined on the positive and negative real axes, where it satisfies all the necessary hypothesis for change of variables, so this also causes no problem.
Abstractly, all this means is that absolutely continuous random variables with values in $\Bbb R^n$ are in 1-1 correspondence with $L^1$ $1$-densities on $\Bbb R^n$ with total integral = 1, the bijection being $X \leftrightarrow f_X(x) |dx|$. This changes correctly according to transformation rules, because if $X$ corresponds to $\omega = f_X(x) |dx|$ then $g(X)$ corresponds to $(g^{-1})^*\omega = f_X(g^{-1}(x)) |\det dg^{-1}(x)| |dx|$
You're pulling back by $g^{-1}$, so maybe you'd want to say it's a contravariant correspondence.

11:33 PM
@BalarkaSen I entered planning to dunk on some kids
turns out its abandoned
why did you go there

11:51 PM
@RyanUnger whim