Mathematics - 2021-04-10 (page 1 of 3)

dc3rd

00:00

We're using the definition of: every neighborhood of a point $a$ contains both points in $S$ and points not in $S$

Ted Shifrin

I use frontier, not boundary. Because students get insanely confused when we get to boundaries of manifolds with boundary.

anakhro

heh

dc3rd

I vividly remember having my head spining in circles when I dealt with the boundary definition copper is talking about the first time

Edward Evans

@HereToRelax I've heard Grand Tour isn't very cool

or whatever it's called

Ted Shifrin

@copper For submanifolds with or without boundary, everything is boundary points. So frontier is just way less confusing.

I am not without reason for my madness.

copper.hat

00:07

@TedShifrin understandable.

@dc3rd using that definition you can see that every point is in exactly one of the interior of $S$, the frontier of $S$ or the exterior of $S$. Since the first & last are open, the remainder must be closed.

dc3rd

visually i see what is happening, in terms of shrinking the size of $\epsilon$ for all neighbourhoods around all my boundary points will give me convergent sequences. Just the solid logic of explaining it is what is missing from me.

if I were to use that definition.

Do we explicitly use these ideas in the upcoming sections @TedShifrin ? I ask because when I did multivariable previously I didn't see these ideas pop up explicitly too much. THey kinda just sat in the background except for defining the usual differentiation and integration ideas, but we didin't use frontier points, limit points, closures, etc too explicitly. I've always wanted to see it in action

Ted Shifrin

00:23

This is not a calculus course. This is a multivariable analysis course. If you study the proofs, they will be used. You will be doing proofs with compactness and integrability ...

Not to mention inverse/implicit function theorems ...

dc3rd

THe course I did was a "middle ground" it wasn't all calculus and wasn't full analysis. We used Folland's "Advanced Calculus"

it was the course for the "inbetweeners" who were not full math specialists and also not math minors.

Ted Shifrin

If you learned that course, you should know most of mine. Hmmm .... Or it could have been taught watered. Down.

dc3rd

"learned" is a stretch..............memorized definitions, theorems, and proofs?.......yes.

but that was my experience because of the weak foundations. But I do recognize all the content you're teaching

also they really didn't do any of the geometry the way you do.

This comes back to the ability to "teach" that you have talked about

HereToRelax

01:04

$$\Demarois$$

HereToRelax

01:19

this person is a legend math.stackexchange.com/a/2353752/33907

apparently someone even tried to post an answer on a sock puppet account to prevent the bounty and got owned

Ted Shifrin

Legend depends on one's universe, evidently

HereToRelax

I thought it was funny :'(

Ted Shifrin

I guess I didn't pay enough attention.

dc3rd

02:33

Perfect timing if you are bored @TedShifrin . I either just finished 9a, or I mixed the results up. What do you think:

i) True/ False: all interior points of $\bar{S}$ are points of $S$?

Claim: False.

Pf: Let $\bar{S} = [a,b]$. Then $b \in S^{f}$. Take $\delta$ nbhds for $b \in S$. If we take $\frac{\delta}{2}$ nbhds we can have $b \in \bar{S}$.

ii) True/ False: If $S$ is open, then all interior points of $\bar{S}$ are points of $S$

Claim: True.

pf: Suppose towards contradiction that there exists interior points $a \in \bar{S}$ such that $a \notin S$. Then $a \in S^{F}$. This means f…

Ted Shifrin

I don't follow a at all. b is wrong.

dc3rd

back to the drawing board...

For a) what I wanted to argue was that the frontier points are also interior points of $\bar{S}$ specifically.

Rover

@robjohn yes, then ...?

Ted Shifrin

@dc3rd check definitioms?

dc3rd

So whatever $\delta$ I have dictating the nbhd around a frontier point of $S$, there exists a smaller $\delta$, this one I said $\frac{\delta}{2}$ that can be chosen to make every nbhd of a frontier point with respect to $\bar{S}$ be fully contained in $\bar{S}$

Ted Shifrin

02:47

Nope.

dc3rd

Alright. I'll go back and read the definitions and proabably check in with you tomorrow about it if you're not around tonight.

Ted Shifrin

The definition of frontier point says nope.

user2236

Has anyone heard about this announcement from the CDC?

> Findings of these studies suggest that the risk of SARS-CoV-2 infection via the fomite transmission route is low, and generally less than 1 in 10,000, which means that each contact with a contaminated surface has less than a 1 in 10,000 chance of causing an infection 7, 8, 9.

Ted Shifrin

It's clearly been about aerosol ... for months. What’s your point?

user2236

is that a "yes", sir?

Ted Shifrin

02:51

I don't know any new news, and I have no idea what 7,8,9 is there.

user2236

sorry, those are the study references

[7], [8], [9]; I guess I'll have to read them

Ted Shifrin

As I say, for months it's been known that it's aerosol transmission that is the grave danger — hence the importance of masks. Hand-washing continues to be a precaution.

user2236

"1 in 10,000 after contact" struck me as odd

Ted Shifrin

Surfaces are not a big deal.

Why odd?

user2236

Thanks for your attention sir.

dc3rd

03:05

Just unraveling the definition a bit more I see what you're getting at with respect to i). To explain it informally. If I take an interior point $a \in \bar{S}$ then there exists a nbhd fully contained in $\bar{S}$. Equivalently it means $a \in S$ or $a \in S^{F}$. We are only concerned with the $S^{F}$ case. If $a \in S^{F}$ then by definition every nbhd of $a$ will contian points not in $S$, negating the definition of an interior point

....... So then that means i) is true.....just have to write it out

Ted Shifrin

03:24

No, it's still not correct. You're not quite catching the subtlety yet. Think about more examples.

dc3rd

Alright will do

The subtlety yo're hinting at has to do with the interior of the set of frontier points I gather.

Ted Shifrin

Not quite.

dc3rd

hmmmm....interesting.

Ted Shifrin

I don't actually know what you mean by that.

dc3rd

Well I was thinking about the set of frontier points exclusively and disregarded the point of $S$ because I thought it was trivial.

Ted Shifrin

03:39

That seems correct, but you went the wrong way down the road.

copper.hat

@dc3rd Pick a point $x$. Then exactly one of the following three things is true: 1. there is an open set containing $x$ that is entirely in $S$. 2. there is an open set containing $x$ that is entirely in $S^c$. 3. neither 1 nor 2 hold.
Now note that if 3. holds, then no open set containing $x$ can be a subset of $S$ or a subset of $S^c$. In other words, any open set containing $x$ must contain points of both $S$ and $S^c$.

Sorry, I've been saving that since before I started dinner.

Ted Shifrin

The issue is to imagine different scenarios with where frontier points are located. Let's not give it away.

StudySmarterNotHarder

@geocalc33 there?

copper.hat

The room is surprisingly lively despite its dearth of tools. I think that is due to you @Ted.

Ted Shifrin

03:56

Is that good or bad?

And whom are you calling tools?

StudySmarterNotHarder

@copper.hat yeah, who da tools ?

copper.hat

:-)

StudySmarterNotHarder

We all can be tools of the prolific spread of mathematics I guess

Ted Shifrin

Tool is also slang.

Perhaps dated slang.

dc3rd

nah it's still relevant. prbably has greater emphasis now because of its age as a term

Ted Shifrin

04:01

As slang, it's rather derogatory. Whence my original question.

StudySmarterNotHarder

Well, if it's slang it could mean 1 of 10 things, and 80% of us will fall into a tool category

Therefore, I think it can be replaced with "people"

Ted Shifrin

Whence my original question.

copper.hat

i just wanted to use the word dearth.

dearth vader

Ted Shifrin

LOL ... or death

StudySmarterNotHarder

dear'th

copper.hat

04:04

a few years ago my daughter gave me a present of "vader's little princess"

StudySmarterNotHarder

He used a bad word
Dearth of t-word, could mean stuff
But who cares, time's little

a Haiku I composed

zacts

I might have some questions in just a bit. I'm working through this #-theory book at the moment.

copper.hat

how did hash become number?

StudySmarterNotHarder

lol

LOL, another victim of the primes

Leonhard Euler

@zacts burton?

zacts

04:06

@LeonhardEuler I'm looking at that and another text too.

Ted Shifrin

# has always been number!

StudySmarterNotHarder

@zacts what's the q?

zacts

illustratedtheoryofnumbers.com

[#] looks like a rubik's cube to me

StudySmarterNotHarder

Math is indeed an infinite-dimensional rubik's cube phenomenon

Just as is coding

zacts

@StudySmarterNotHarder well, I'm kind of stuck with some of the problems. maybe my first question could be what's a good study strategy for these texts? Also, might analysis be a useful way for me to get started before these texts, like with Tso's Analysis for example.

copper.hat

04:08

i think i have only seen it used for number in the us, at least a few decades ago.

StudySmarterNotHarder

@zacts what part do you first get stuck on in the book?

zacts

@copper.hat what is your etymological meaning of #?

copper.hat

the interweb is homogenising.

zacts

@StudySmarterNotHarder I'll link a particular problem

StudySmarterNotHarder

Seems also like you need to know some Python from that page

zacts

04:09

@StudySmarterNotHarder no python is needed for the actual text of an illustrated theory

that's like extra supplement for the text

StudySmarterNotHarder

Oh, cool, either way I could teach Python 3.x / SymPy coding if that's what you wanted

copper.hat

i could easily have forgotten, but i believe it was used to indicate a number as a label.

as opposed to having some order. similar to a modern SKU

zacts

#:

StudySmarterNotHarder

$\# A$ is an alternative to $|A|$ when $A$ is a set

copper.hat

that is where the cs usage came form

zacts

04:11

@copper.hat ah cool

Ok, let me link a question from this text by burton

StudySmarterNotHarder

k

copper.hat

it is called an octothorpe

zacts

Prove the following facts concerning triangular numbers: (a) A number is triangular iff it is of the form n(n+1)/2 for some n>=1.

StudySmarterNotHarder

Another way to phrase that is they want you to prove that $T(n) = \dfrac{n(n+1)}{2}$

zacts

The information given is: "This led the ancient Greeks to call a number /triangular/ if it is the sum of consecutive integers, beginning with 1."

copper.hat

04:14

is that a definition?

StudySmarterNotHarder

where $T$ is the sequence of triangular numbers

It's actually a very well-tought proof, what are you having trouble with? It's an obvious candidate for inductive-style proof because $n + 1$ occurs directly within it and that's how induction proceeds.

Prove it first for $n = 1$

zacts

I think my confusion was with the definition for /triangular/.

StudySmarterNotHarder

$1(1 + 1)/2 = 1$, done

Oh, cool

zacts

triangular means "The sum of consecutive integers, beginning with 1"

so if I can prove that n(n+1)/2 is the sum of consecutive integers, beginning with 1,, I have proven it's triangular, right?

iff that is

StudySmarterNotHarder

I guess because you can form a "pyramid" with 1 object at the top, 2 objects on the row below that, 3 on the row below it and so on, and the sum of all row counts is equal to the area approximately (as $n$ grows) of a triangle.

copper.hat

04:17

Show $T(1) = 1$ and $T(n+1) = n+1+T(n)$.

StudySmarterNotHarder

If part is proof the formula $T(n) = \dfrac{n (n+1)}{2}$, and only if would be simply by definition I think

zacts

The integer n is a triangular number iff 8n+1 is a perfect square.

I wonder if I should start with a different text first.

I also am looking at this Analysis I text by Tso.

dc3rd

Ok either I'm thinking too hard or I'm on to something. But I've been sitting here thinking about the statement of

"either $a \in \bar{S}$ or $a \in \bar{S}^{c}$ or $a \in \bar{S}^{F}$"........should I unravel the idea from there?

StudySmarterNotHarder

For that you're asked to solve: $8 T(n) + 1 = m^2$ for $m$, for any given $n$. And the converse part is for all $m$ such that $m^2 = 8 k + 1$, we have that $k = T(n)$ for some $n$.

zacts

but let me post a cool looking question from the other #-theory (pardon my use of #) book.

StudySmarterNotHarder

04:23

Nice, let's hear it

*see it

copper.hat

@dc3rd i'm not sure what you are trying to do.

StudySmarterNotHarder

Either $a \in A$ or $a \notin A$ for all objects $a$ in the universe of thought, for any given set $A$.

copper.hat

try a few examples. what is the frontier of $\mathbb{R}$?

zacts

Draw a spiral to demonstrate that 100 = 10 + 2(9) + 2(8) + 2(7) + 2(6) + 2(5) + 2(4) + 2(3) +2(2) + 2(1).

StudySmarterNotHarder

What does the spiral look like?

zacts

04:26

I don't know.

StudySmarterNotHarder

What's the general pattern they are implying?

zacts

it's got to be stacked somehow

like stacked dots

StudySmarterNotHarder

Then I would try it with a smaller number than 100 maybe

zacts

ok

yeah

StudySmarterNotHarder

25 maybe

zacts

04:27

that's kind of what is going on in the ch for sure

I don't think this would be too difficult now that you mention it.

let's see if I can sketch a pic for that.

StudySmarterNotHarder

25 = 5 + 2(4) + 2(3) + 2(2) + 2(1)

Look at each term, can you tell me the general formula for a any square $n^2$, $n \in \Bbb{N}$?

dc3rd

That's a good question @copper.hat ....I'm drawing it now and I'm asking what IS the frontier of $\mathbb{R}$..only thing that comes to mind is $-\infty$ and $+\infty$ and those don't mean much.

StudySmarterNotHarder

The frontier of $\Bbb{R}$ is the great wide open $\Bbb{C}$-plane, son!

J/k

zacts

@StudySmarterNotHarder n^2 = 2n(n+1)/2

wait

dc3rd

I was wondering if it was $\mathbb{C}$ or not.

StudySmarterNotHarder

04:31

I was speaking poetically though

copper.hat

Lets stick with reals for now.

It is not the complex plane.

StudySmarterNotHarder

$n^2 = n + ?$

copper.hat

what points can be approached from the complement?

zacts

sorry just a sec..

StudySmarterNotHarder

Use \sum_{k = 1}^{n-1} ?$ notation

dc3rd

04:33

if a point is $x \notin \mathbb{R}$....hmmmm....I'm truly stumped

would we have to include higher dimensions?

copper.hat

Noooooooo

$\mathbb{R}$ is the whole universe here.

dc3rd

Well maybe I shouldn't search too hard because all I got is the empty set beyond $\mathbb{R}$

copper.hat

exactly the complement is the empty set

StudySmarterNotHarder

@zacts I'm not sure about what spiral it should be

copper.hat

or look at it this way, for any $x$ you can find a open set containing $x$ that is a subset of $\mathbb{R}$.

dc3rd

04:36

but the empty set has no points in it, so how can I have a frontier point at "the end" of $\mathbb{R}$?

copper.hat

if $x$ was a frontier point then any open set containing $x$ must contain points in $\mathbb{R}$ and $\mathbb{R}^c$.

StudySmarterNotHarder

Peas porridge hot, peas porridge cold

copper.hat

however, there are no points in $\mathbb{R}^c$.

dc3rd

ha.....which it cannot because there are no points in $\mathbb{R}^{c}$

copper.hat

from which we conclude that the frontier of $\mathbb{R}$ is ....

zacts

04:38

imgur.com/zR3piku

copper.hat

remember that the frontier is a set.

zacts

^ I think it's got to be something like this stacked dots idea

dc3rd

ooooohhhh..............the set is empty

copper.hat

exactly

how about the rationals as a subset of the reals.

what points $x$ can be approached arbitrarily closely by rationals and irrationals?

dc3rd

See the rationals I used as an example of a frontier set.

copper.hat

04:41

so what is the frontier?

dc3rd

sorry I used the integers

copper.hat

what real numbers can be approached arbitrarily closely by rationals?

dc3rd

the integers would serve as a frontier set

copper.hat

not sure how you're getting that.

in a metric space, a point is in the frontier if (and only if) it can be approached arbitrarily closely from the set and its complement.

the number ${1 \over 2}$ is a rational that is far away from the integers.

StudySmarterNotHarder

@zacts I think you take a square, say $n^2 =25$ and draw an actual square of 5 x 5 units

Cross out the top row (5)

and fit in $2(k)$ for each $k \lt 5$

in a spiral-like pattern

dc3rd

04:46

well to your question of what real numbers $x$ can be approached by rationals. I would say the irrationals.

zacts

@StudySmarterNotHarder would it be $n^{2} = 2(n-1)n/2$?

oops

copper.hat

that is not all

dc3rd

$0$ as well

copper.hat

well, any rational

StudySmarterNotHarder

$n^2 = n + \sum_{k = 1}^{n-1} 2k = n + 2 \sum_{k=1}^{n-1} k = n + 2 T(n-1)$

zacts

04:47

$n^{2} = n + \frac{2(n-1)n}{2}$?

StudySmarterNotHarder

yes

zacts

I left in the 2 there to reflect the grouping in the original problem

StudySmarterNotHarder

But the spiral they ask for is starting with an $n \times n$ square.

dc3rd

and then vice versa for the irrationals towards rationals as well.

copper.hat

@dc3rd given any real $x$, and an open set $U$ containing $x$. Does $U$ contain rationals?

zacts

04:49

@StudySmarterNotHarder oh, so I'm drawing a spiral within the square?

like draw a line within the dots kind of idea?

StudySmarterNotHarder

But the spiral for case $n$ has in it the spiral for case $n - 1$ which has in it the spiral for case $n - 2$, and so on down to $1$.

I would use grid squares

Where each grid box is a unit in area

dc3rd

Yes it does.

copper.hat

Does $U$ contain irrationals?

dc3rd

Definitely does

copper.hat

So what is the boundary of the rationals?

dc3rd

04:52

Was just going to type it too.....the boundary is the reals

copper.hat

correct. good.

StudySmarterNotHarder

551133
552233
552233
554444
554444

copper.hat

what is the boundary of the irrationals?

StudySmarterNotHarder

@zacts where the labeling just reflects what it index it corresponds to

copper.hat

sorry i meant frontier. old habits die hard.

dc3rd

04:56

boundary is the rationals.

Ted Shifrin

Nope.

dc3rd

lol....ted with the RKO out of nowhere......

copper.hat

what real numbers can be approached arbitrarily closely by irrationals?

i'm going to ask the same questions again...

Ted Shifrin

Don't worry: I'll disappear as quickly as I appeared.

zacts

@StudySmarterNotHarder is the spiral formed from the actual dots themselves within the grid?

StudySmarterNotHarder

04:58

The spiral is centered approximately at $1 \ 1$

You then double 2 to get: $1 \ 1 \\ 2 \ 2 \\ 2 \ 2$

That always gives you a height or width that is $3, 4, 5, \dots$

The next is 3 so:
1 1 3 3
2 2 3 3
2 2 3 3

The point is that it's always of the form $k(k-1)$

dc3rd

real numbers that can be approached by irrationals are the irrationals

copper.hat

that is a different answer to the same question when we were looking for the boundary of the rationals.

can 2 be approached by irrationals?

dc3rd

yes it can

copper.hat

so, what real numbers can be approached arbitrarily closely by irrationals?

dc3rd

the integers

but then the rationals contain the integers.....

copper.hat

05:05

lets back up a bit

pick any real $x$. Pick any open set $U$ containing $x$.

Does $U$ contain irrationals?

dc3rd

was just doing that too.... but yes $U$ contains irrationals

copper.hat

Does $U$ contain rationals?

zacts

@StudySmarterNotHarder thanks. I'm still trying to visualize this, but that looks like a cool solution.

StudySmarterNotHarder

yw

dc3rd

Yes $U$ contains rationals.

so the reals would serve as a boundary for the irrationals...

given that the irrationals are viewed as a subset of the reals.....very important

copper.hat

05:12

correct.

note that the boundary of the rationals and the boundary of its complement (the irrationals) is the same.

robjohn

@TedShifrin Poof!

copper.hat

i mean frontier

robjohn

@copper.hat depending on the book you use

dc3rd

the subtlety in this is keeping note of what is a subset of what. I understand what you mean by boundary, no worries

copper.hat

what is the frontier of $(-\infty,0]$ as a subset of the reals?

@robjohn Ted wants me to use frontier in current discussion so i am complying :-)

zacts

05:14

@StudySmarterNotHarder just to be clear, but was my solution to the $n^2$ problem correct?

robjohn

@copper.hat Yes, master.

copper.hat

@dc3rd It is important, since complements are involved.

i am very obedient :-)

zacts

$n^2 = n + 2[\frac{(n-1)n}{2}]$

StudySmarterNotHarder

@zacts yes

zacts

ok, thanks

robjohn

05:16

@zacts $n^2=2\binom{n}{2}+\binom{n}{1}$

dc3rd

the frontier would be $(0, \infty)

copper.hat

nope

what $x$s can be approached from inside and outside the set?

robjohn

@dc3rd are you using the definition $\partial S=\overline{S}\cap\overline{S^C}$?

copper.hat

for any $y>0$ i can find some $r>0$ such that $(y-r,y+r) \subset (0, \infty)= (-\infty, 0]^c$.

dc3rd

no closures on them @robjohn

robjohn

05:18

@dc3rd then that would be empty

zacts

@StudySmarterNotHarder @robjohn thanks

copper.hat

@dc3rd loosely, for 'nice' sets, the frontier is the 'edge' of the set.

what is the edge of $(-\infty, 0]$ ? In the context of the reals?

robjohn

@copper.hat that would seem to be what I wrote...

dc3rd

in this case then $0$ is the frontier

StudySmarterNotHarder

@zacts in your proof you have to state that $\sum_{k = 1}^{n-1}k = T(n-1)$

robjohn

05:20

@copper.hat $\partial S=\overline{S}\cap\overline{S^C}$

zacts

@robjohn would you mind explaining in a nutshell how that notation works?

copper.hat

@robjohn i think @dc3rd is using the defn that $x$ is in the frontier iff any open set containing $x$ contains points in the set and its complement.

robjohn

@zacts which notation?

copper.hat

which is, of course, equivalent,

StudySmarterNotHarder

{ n \choose r }

copper.hat

05:21

and the closure definition is one i prefer.

zacts

the bionomial coefficient notation parenthetical (n/2) (n/1)?

copper.hat

binom{n}{m}

with a slash

backslash

zacts

$binom{n}{m}$

dc3rd

verbatim the definition is: "$a \in \mathbb{R}^{n}$ is a frontier point of S if every neighbourhood of $a$ contains both points in $S$ and points not in $S$"

zacts

$\binom{n}{m}$

robjohn

05:22

@zacts right-click on the mathjax?

zacts

ok

enthu

@TedShifrin I have complex shapes of geometries which makes it hard to integrate over the area, I am thinking to use the theory to make things more analytical and reduce integrations. do not know though how things may work or succeed.

robjohn

\binom{n}{m}

zacts

@robjohn no, can you conceptually explain how bionomial coefficients work in a nutshell, not how to write it in $\latex$.

robjohn

ah, okay

zacts

05:23

Although, I do apologize as I'm still learning LaTeX at the moment as well

I do have another venue for a practice pad too which I'll use.

copper.hat

@dc3rd in my example. pick 3 different classes of points. $x<0$ $x=0$ and $x>0$.

robjohn

@zacts They are the coefficients of $(1+x)^n=\sum\limits_{k=0}^n\binom{n}{k}x^k$

copper.hat

for each class. ask if it can be 'approached' from in the set and from outside the set.

dc3rd

well I was typing something, but I'll save it. I think I'm going to conclude the same thing, but it is worth the practice.

robjohn

@zacts They can also be computed in Pascal's Triangle

copper.hat

05:26

@dc3rd you need to work examples involving intervals open closed half inifinite and become comfortable with the reasoning.

love_sodam

Can anyone describe a simply connected covering space of a 2-sphere with a circle intersects with two points?

robjohn

@zacts also $\binom{n}{k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)\cdots(n-k+1)}{k!}$

copper.hat

@dc3rd as i mentioned, in 'nice' cases the frontier is what you might expect the 'edge' of the set to be.

i should not have started with the rationals.

love_sodam

I don't know how to describe when a portion of circle is inside of sphere

copper.hat

a not so great circle?

robjohn

05:28

@zacts I am not sure what you mean by "how they work"

dc3rd

Got it. with regards to 'approached', I take it you mean that it can "eventually" get extremely close to the point?

zacts

@robjohn how would you think of that notation conceptually within the context of the $n^2$ problem, perhaps in terms of how things are grouped together? if that makes sense. I mean you wouldn't be thinking in terms of the factorial formula given above right?

copper.hat

@dc3rd well, more precisely, any open set contains points from the set and its complement.

robjohn

@zacts No. I was simply considering exactly the formula you stated: $\frac n1+2\frac{n(n-1)}2$

but thinking of it as binomial coefficients allows it to be manipulated using things used to manipulate the binomial coefficients

such as the hockey-stick identity

Or Vandermonde's Identity

zacts

what would $\binom{n}{3}$ look like?

robjohn

05:34

@zacts $\frac{n(n-1)(n-2)}{3!}$

8 mins ago, by robjohn

@zacts also $\binom{n}{k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)\cdots(n-k+1)}{k!}$

zacts

so, would $\binom{n}{4} = \frac{n(n-1)(n-2)(n-3)}{4!}$

ok

robjohn

@zacts yes

in $\binom{n}{k}$ I only consider $k$ being a non-negative integer. $n$ can be any real (or complex even)

zacts

so you were directly notating my original solution to the $n^2$ problem in terms of binomial coefficient notation, which then allows you to use operations specific to binomial coefficients.

robjohn

yes

zacts

cool, thanks.

copper.hat

05:37

@zacts there is a nice interpretation as the number of paths to get from the top to a point in Pascal's triangle.

robjohn

@copper.hat did you see my Paschal Triangle?

Mar 29 at 13:52, by robjohn

copper.hat

:-)

robjohn

the number of stripes on each egg is important

Ted Shifrin

@enthu I'd need to see an example to comment further.

copper.hat

very seasonably appropriate

dc3rd

05:40

of course the eggs have tge powers in the stripes

zacts

what do the colors mean?

dc3rd

expand some binomial variables and you will see it

copper.hat

i can't figure out the coloUr mapping

zacts

oh wait are those easter eggs?

robjohn

@zacts the number of colors is the value of the same place in Pascal's Triangle

@zacts yes

@copper.hat The colors are random, the number of colors is important

copper.hat

05:41

@dc3rd what is the boundary of $\emptyset$ in the reals?

@robjohn i got the number part, i thought there was another map

dc3rd

he was pop quizzing me @robjohn

copper.hat

i'm quizzing dc3rd

robjohn

sorry

copper.hat

part of the fun :-0

dc3rd

lol.....too late to delete the answer.

copper.hat

05:43

note that the empty set is the complement of the reals in the reals.

zacts

I'm kind of looking for a text to latch onto this kind of idea.

copper.hat

in general the frontier of a set and its complement is the same.

dc3rd

I think that was actually the next question I have to do in the text

zacts

I'm also looking at this text, Analysis I by Tso.

robjohn

@zacts my favorite text for discrete math is Concrete Mathematics

zacts

05:47

I have that book too, and I'm looking at that, but I wonder if I'm ready for it.

this discrete type of mathematics definitely heavily interests me

copper.hat

i like flamboyant math.

i prefer short dense books.

robjohn

@copper.hat (shhh!)

zacts

I think I was confused with the [p prime] notation.

Iverson's convention

robjohn

@zacts The Iverson brackets?

yes

zacts

yeah

this was like a year ago I tried to dive into it a bit

I'm looking at the book now

robjohn

05:51

@zacts I use those in several of my answers

66 to be exact :-)

zacts

@robjohn hum... I wonder what text I might dive into at the moment.

copper.hat

the pleasures of counting

probably not what you are looking for, but a nice read.

Lucas Henrique

hey chat, good evening

zacts

Would you say that understanding some real analysis would be a prerequisite for Concrete Mathematics, or could one just dive into the book?

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