Mathematics

8:00 PM

So $f_* : \pi_1(S^1, 1) \to \pi_1(S^1, 1)$ passes down to another homomorphism from $\mathbb{Z} \to \mathbb{Z}$ since $\pi_1(S^1, 1) \cong \mathbb{Z}$ and then that homomorphism is multiplication by some $m \in \mathbb{Z}$ and that $m$ we call the degree of $f$

Daminark

Yup

Perturbative

Cool cool, thanks @Dami

Akiva Weinberger

And $\pi_n(S^n)$ is also $\Bbb Z$ so you have degrees of maps from $S^n$ to $S^n$ as well

Hm, I wonder what happens if you use other homotopy groups of spheres

Daminark

Usually you see degree defined in terms of homology. Now, it turns out that there's this guy called the Hurewicz map, which in this case maps $\pi_1(X) \to H_1(X)$. In this case that guy is an iso so it's all good

Akiva Weinberger

Yeah, $H_n(S^n)$ is also $\Bbb Z$ and is easier to work with

Perturbative

8:03 PM

I was just reading about the Hurewicz map @Dami

Akiva Weinberger

You know what I need to learn about at some point? Topological dimension

(Small and large inductive dimensions and Lebesgue covering dimension)

Perturbative

Is the degree defined in terms of homology more general? Because then we don't need to worry about pointed maps and basepoints and all that jazz

Akiva Weinberger

It works without basepoints

Daminark

Yeah, this is nice because it generalizes

So, it turns out a closed orientable manifold has top homology $\mathbb{Z}$

Akiva Weinberger

$\pi_1$ doesn't need basepoints, really, as long as it's connected

Daminark

8:05 PM

@Akiva well, to talk about degree you need to know stuff about induced maps, and I think it's possible that a change of basepoint map can sometimes flip sign. Probably not that big of a deal anyway but maybe

Akiva Weinberger

The degree is also basically the number of preimages of each point, counted with some sort of multiplicity or something

Érico Melo Silva

sup nerdos

Daminark

Yeah, that's the other version. If you have a smooth map between closed orientable manifolds, then the preimage of a regular value is gonna be a finite set. If you count each point with either + or - depending on the sign of the determinant of the jacobian, you get the degree.

Akiva Weinberger

Points could be +2 as well, can't they?

Like $z\mapsto z^2$ on the Riemann sphere

(The suspension of the usual degree-2 map on the circle)

The poles get weird

Unrelatedly

Cuisenaire rods

Cuisenaire rods are mathematics learning aids for students that provide an enactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891–1975), who called the rods réglettes. According to Gattegno, "Georges Cuisenaire showed in the early fifties that students who had been taught traditionally, and were rated ‘weak’, took huge strides when...

Érico Melo Silva

@Daminark have ur letter writers gotten to it yet

Daminark

8:13 PM

I haven't really sent in the application forms yet, way too much going on in classes

Probably over Thanksgiving I'll do so

Érico Melo Silva

schlaggy just sent em in for me

so that's my last round done

Daminark

Excellent

Érico Melo Silva

highly nervous

Astyx

8:27 PM

Hi chat

Julius

8:39 PM

Hi. I have $E[|A_n / B_n - C|]$ where $A_n$ and $B_n$ are random variables, and $C$ is a constant. I also know that $B_n\to B=const$ in probability. Any ideas how to get rid of $B_n$ from the denominator by bounding the expression from above?

user193319

9:01 PM

@AkivaWeinberger Would you mind expanding on that a little bit? The function $f$ is a map from $\Bbb{R}^p$ to $\Bbb{R}^q$. If I'm not mistaken, the mean value theorem doesn't hold for such functions.

Jake S

9:18 PM

Welp, I just posted a question on quadric surfaces, if anyone has time and feels like helping me out. Thanks in advance to any kind hearts and no problem to anyone who doesn't have time or energy for the task!

FuzzyPixelz

9:36 PM

In case you are interested, I'm trying to solve (d), is it helpful to think about the sequence as $a_n \in \{\frac{m}{n}| 0 \lt m \lt n \}$ ?

James Groon

If I put |f(x)| = x , what are all the possible outcomes for f(x) ? Is it x and -x only or is there an infinite solutions ?

Leaky Nun

depending on whether you want f to be continuous

James Groon

So in general , there is infinite amount ?

FuzzyPixelz

Yes, your function could equal $x$ or $-x$ on any point and you have infinitely many points to play with, loosely speaking. I'm not sure how to prove it though.

James Groon

9:52 PM

Thanks man , i hope someone answers your question too :)

Jake S

Hmm, not sure if anyone will have time to answer the lengthier description of my question at the moment, but how do I find the reduced equation of a quadratic form given an invertible matrix $Q$ and its diagonal form?

James Groon

I would say that a can be any rational number in [0,1] cuz for any n/m , n<m you will have n/m , 2n/2m , 3n/3m ... desired subsequence.

@FuzzyPixelz

PabloZ392

Hi :) I have three random variables :1 The result of rolling the dice. 2. The number of reverse(tails) in three coin toss. 3. $\lfloor x^2 \rfloor$ where $x \in [-2,2]$ . Can we describe above random variables on the one, standard probability space? For example $\Omega =[0,1]$

Transcript for

$Mathematics$ Mathematics