Mathematics - 2022-01-22

love_sodam

00:30

Adding syrup to my coffee was a huge mistake.

Ted Shifrin

Maple syrup?

love_sodam

I don't usually put syrup in it, so I don't know what syrup it is, but it said "café syrup".

Ted Shifrin

Never heard of that! Plenty of sugar. Which I hate.

robjohn

04:23

@JavierMorenoSepena What is to be shown about this distribution?

@TedShifrin you hate sugar?

It seems I am talking to ghosts

leslie townes

boo

robjohn

eek! you scared me

Rithaniel

leslie with the scare tactics

leslie townes

... boo!

copper.hat

went winetasting. got 6 bottles of 1 2012 hafner. will attempt to drown the bad week.

robjohn

04:34

all 6 bottles?

copper.hat

:-). no, just kidding. they will stay in the basement for while until we visit or have visitors.

i am too tired to drink tonight :-)

Ted Shifrin

04:58

@robjohn In coffee, decidedly!

leslie townes

simple syrup has its uses, but not coffee.

we need to stop agreeing, ted.

pepsi or coke?

copper.hat

i used to add so much sugar to my tea that it was super saturated.

diet coke of course

gale force warning here tonight

Ted Shifrin

@copper.hat Shocking!

copper.hat

i drink tea with no sugar and skim milk now. the threat of cholesterol meds was enough

Ted Shifrin

Oh, I’ve been on those and 2 different BP meds for almost 20 years.

copper.hat

05:03

this was when i used to exercise significantly, partly because of the 3 gallons of full fat milk.

Ted Shifrin

Gale force winds in Albany?

copper.hat

bay area. doesn't sound like much outside

Ted Shifrin

I’ve drunk only skim milk since early childhood.

leslie townes

once you go off whole milk it tastes disgusting. like butter, or like it's gone off. and it hasn't, it's just too fatty.

only skim at my house.

copper.hat

we used to drink fresh milk growing up, sometimes directly from the cow (well, run through the separator and cooled).

Ted Shifrin

05:05

I love good butter on good fresh bread or good English muffins ….

copper.hat

freshly baked bread is hard to beat

leslie townes

if you don't eat dairy for a while, it can be hard to bear. just the smell of it can be nauseating.

but i had blue cheese on crackers last week.

copper.hat

i suspect that, other than fasting days, i had dairy every day of my life.

trader joes has not had cotswold cheese lately which is killing me

leslie townes

i've gone through weird diets over the last year. there were about three months where i did not eat dairy, or meat, or caffeine, or alcohol.

i'm firmly back on all of those things, but if you stop for a while, you can definitely smell dairy on people.

copper.hat

realistically i cannot imagine life without dairy

i spend some months in China, i think it probably happened then

leslie townes

05:12

when i'd quit it, i realized, this is delicious, i'm being dumb. many people who love this stuff cannot tolerate it for some health reason, and yet i can.

hence the snacks with the blue cheese.

i don't think that some of the ethical concerns that attach to meat eating also attach to dairy farming. i grew up near several dairy farms and the cows seemed delighted.

and i still eat meat, too.

copper.hat

i don't have ethical issues with eating animals. i don't like seeing animals mistreated of course.

Ted Shifrin

How about my cat, who mistreats me?

copper.hat

your cat should have ethical issues

Ted Shifrin

Indeed.

Koro

copper: greenqueen.com.hk/octopuses-lobsters-crabs-sentient/….

copper.hat

05:56

We need to focus on treating other people properly first...

CroCo

Hi there, I'm trying to find the kinematic singularities of a robot. The book says these postures occurs when the robot resists motion which corresponds to the nullity of the jacobian matrix. I've computed the nullity but I'm not able to find such angles. See the following picture.

Is it possible from the nullity and the matrix's transpose to find such angles?

Ted Shifrin

Places where the Jacobian drops rank, not necessarily nullspace!

CroCo

@TedShifrin how to check when the rank drops?

Ted Shifrin

Likely Non-manifold points of its configuration space is what I said

So max rank is what?

CroCo

3

for this example

Ted Shifrin

06:10

Right. So when does it drop to 2?

Think about columns.

CroCo

when at least one column becomes dependent, I think.

Ted Shifrin

But look at how that can happen here.

CroCo

to be honest, I'm not sure.

Ted Shifrin

Throw away the zero columns. There are three columns left. Can you use the column of 1s to get a nontrivial linear combination?

Equal to 0, I mean, of course.

CroCo

one moment.

I think that is not possible.

You mean this, right?

copper.hat

06:23

Presumably the $L_k>0$ and $s,c$s are $\sin, \cos$?

CroCo

yes

copper.hat

One way it can drop rank is if the last two columns are multiples of each other.

CroCo

@copper.hat true but in this example at which angles this occurs? I'm trying to see it but I couldn't

Ted Shifrin

That’s what I wanted Croco to figure out.

CroCo

or may be at 45

Ted Shifrin

06:28

No, constant multiple allowed!

But the same constant on every row.

What ratio?

It looks like a mess, though.

Oh, do row operations !

CroCo

row echolen form?

Ted Shifrin

Sure. Just the two columns we’re talking about.

CroCo

@TedShifrin Matlab gives this

[1, 0]
[0, 1]
[0, 0]
[0, 0]

Ted Shifrin

You can’t use matlab.

You have to do it by hand.

CroCo

sure

copper.hat

06:44

perhaps you could think in terms of $\tan (\theta_1), \tan(\theta_1+\theta_2),...$?

gn

Ted Shifrin

Nighty night!

CroCo

@TedShifrin I think this is it.

what next?

love_sodam

07:04

Matrix group is interesting although I don't know it well.

Ted Shifrin

You’re not thinking smart, Croco. You’re proceeding mechanically.

Think about using the first row to simplify the others. Then think ratios.

CroCo

@TedShifrin sometimes I can't think out of the box. I admit it.

but I like your way by pushing your students to their limit.

@TedShifrin the first row is zero. what do you mean by simplifying the others.

CroCo

07:37

Now I'm thinking smartly.

@TedShifrin

Do I need to equate ratios with zero so these angles are my target?

Koro

08:40

Let S be a parallelogram not parallel to any coordinate plane. Let S1, S2,S3 be projections of the parallelogram on the three coordinate planes. Then the area of the parallelogram is....?

How do I find this area using surface integrals?

uhoh

09:08

3

Q: Has there ever been a case where someone wished a theorem or important result wasn't named after them? Has it happened more than once?

The current answers to the Academia HNQ Should I ask for permission to name a mathematical theorem? can be loosely paraphrased as "go for it" and "it would be bad if you didn't", e.g. There is no need to ask permission, and mostly likely they will be happy to have the theorem referred to with th...

leslie townes

hah, that's a very interesting question.

i think sometimes there is disjointness between what a person regards as important, and what gets named after them. usually the people have no control over what is named after them. it will be interesting to see the responses.

MethNoob

10:04

I am loser abd mentally retarded person I am worthless piece of poop and I will never be a successful mathematician.

Jakobian

10:16

You don't have to be a mathematician. It doesn't dictate your value either

It's a normal thing to have those kind of feelings when you're trying to push yourself too hard, I guess?

You might try to push forward, but you'll have to change your mentality

leslie townes

it isn't helpful to characterize oneself or any person as "mentally retarded." these phrases say more about the people who employ these terms than the people they are applied to. even if they're the same people.

MethNoob

10:34

Sometimes I feel I am not creative enough.

Can't think outside the box.

Jakobian

10:48

Is that really the issue though

I think, in math, you'll meet lots of situations like that. You'll be stuck without progress. That's normal

But you're saying, you have low value in yourself because of those things. That's abnormal

I'm not saying such things are easy to not get frustrated about

But you shouldn't take it out on yourself, or at least try to

Koro

11:07

How should I parametrize a surface?

I understand that $z=f(x,y)$ can be parametrized as $r(u,v)=(u,v, f(u,v))$.

But what about parametrizing that region of a plane p that gets cut off by a cylinder $x^2+y^2=a^2$?

Here is the question that I am trying to solve: math.stackexchange.com/questions/3250821/…

Since I'm confused at parametrizing the region, here's a solution that I came up with:
$z=a-x-y$ so $z_x=-1=z_y$
The required area $a(S)= \int\int_T \sqrt{1+z_x^2+z_y^2}dxdy$.
What is $T$? $T$ is basically the projection of the plane on the cylinder. So $T$ is the region: {(x,y):x^2+y^2<=a}.

Leaky Nun

@MethNoob what's wrong with never being a successful mathematician?

7

Koro

Using polar coordinates: $a(S)=\int_{0}^{2\pi}\int_{r=0}^a\sqrt 3 r dr d\theta=\sqrt 3\pi a^2$

Is that correct? Thanks.

MethNoob

11:25

Ah I finally solved the problem I am the best

@LeakyNun doesn't it sound regrettable when you spend so much time on math and then you fail to contribute to mathematics before dying?

Leaky Nun

11:42

@MethNoob depends on why you spend time on maths

if you had fun and enjoyed it, then there is nothing to regret about

and also depends on what you mean by "contribute to mathematics"

if you have shared a piece of maths that you like with your friends, you have already contributed to mathematics

@MethNoob the mind can often be trapped in its own negative thoughts and as a result view the world negatively; and the solution is to take a step back and view your thoughts as opinions that you generate, and analyse those opinions as they are

"Most people don't think about things. Thoughts appear in their head and they believe them."

Jakobian

About products of Baire spaces again. Well, I didn't prove it myself, but I found an article by Oxtoby which seems to prove this, and I will probably analyze it.

eudml.org/doc/213586

This one

Jakobian

12:06

He basically shows that a product of arbitrary amount of second-countable Baire spaces is Baire

Alessandro Codenotti

13:37

@Jakobian that's a nice result

For finite products of second countable spaces it is an easy corollary of Kuratowski-Ulam of course

Astyx

anyone know how to reduce the size of an image in a post on main ?

Jakobian

Mhm ^^
According to the article, to prove the arbitrary product case, you first have to deal with the countable product one by using Kuratowski-Ulam repeatedly.

Astyx

Specifically in this answer I would like the first image to be much smaller and - if possible - centered

1

A: How to maximize area bounded by thread and a fixed line?

You are asking the same as maximizing the surface area between the thread and the thread and the diameter, imposing that the thread stays below the half circle. The solution to this problem is the arc circle: indeed we know that the circle has optimal area/perimeter ratio. Consider the circle of...

Jakobian

There's example <img src="https://example.com/sample.png" width="100" height="100"> here stackoverflow.com/editing-help, I think it works the same

Astyx

Ah ok thanks

Jonas

14:05

Is there any "intuitive" relationship (say geometric, for example) between a matrix and its adjugate/adjoint? I have an assignment which asks me to "determine and calculate" the adjugate of a given matrix. I'm confused regarding how I would "determine" it.

Jakobian

Maybe they mean for you to provide the definition

determine usually means the same thing as calculate so maybe they just mean for you to calculate it

Jonas

maybe yes

it's probably best to ask some of my fellow students on monday and see how they understood it :)

robjohn

14:48

@Astyx try <img src="https://i.stack.imgur.com/5lAfH.png" width="350">

or whatever width you like

I did a test edit and it works (I didn't save)

This works even better: <a href="https://i.stack.imgur.com/5lAfH.png"><img src="https://i.stack.imgur.com/5lAfH.png" width="350"></a>

That will let them click on the smaller image and get the full size image

Alessandro Codenotti

@Jakobian that's reasonable, failure if Bairness should be witnessed on a countable product. Iirc if $X^\kappa$ is not Baire, then $X^\omega$ already isn't. Don't quote me on that though, I haven't looked at those things in a while

Odestheory12

16:52

Any idea how to run this applet? mathinsight.org/applet/…

Astyx

17:16

@robjohn Thanks I ended doing this. for some reason Jakobian's example didn't work for me

Jakobian

well, it's not my example, I only googled it

Astyx

I had googled it, but I thought MSE used BBcode and not Markdown

I don't know why it didn't work though -- maybe I made an error when copying it

thanks for your help either way

Hawk

@Jakobian so I think I know what happened. Basically defining smooth functions on aren't defined on open sets can lead to not so well-defined derivatives. I used a one-dimensional case before like f(x) = x on [0,1), but that I found out it was very difficult to see what was wrong. But a good example math.stackexchange.com/questions/2843163/… can be found here when we add 1

more dimension, the problem becomes clearer. So I guess that's why we compare against functions f on arbitrary sets against smooth functions thet are by default defined on open sets

MethNoob

17:55

I don't know if I am asking dumb question but I have blindly believing proof of pyramid is true

The proof was for a pyramid with square base

which is obviously 1/3l^3 for length l

Jakobian

What proof?

MethNoob

but when it is rectangular one can you show me a geometric proof that the volume is 1/3bh where b is base and h is height

nrich.maths.org/1408

I have enough tools to prove it but I think this method doesn't obviously show volume of pyramid with rectangular base

how do you know that the pyramid inside the rectangular prism is exactly 1/3 of it's volume if show that way in the link

when not using calculus

Ted Shifrin

The link you just gave justifies the formula for every rectangular base and triangular base (and from that you can justify any base if you allow limiting processes).

MethNoob

all three pyramid will have different base and different height

Ted Shifrin

In general, you cannot, I believe, fit three congruent shapes inside a box. But you use similarity arguments as the person in that link does.

MethNoob

18:05

oh shoot I didn't read that proof

I read another website

proof

Ted Shifrin

Just shows you should take your own medicine before giving it to us :P

MethNoob

lol

MethNoob

18:33

what's bothering me with this link proof is how they knew what scale factor was without knowing the volume of pyramid with height h and base a^2

scale factor should be (volume of pyramid with height h)/(volume of pyramid with height a)

MethNoob

18:49

but instead if it is treated as very thin slice of rectangular prism

oh now i can see thanks

i hope you enjoy the moosic

I am a very good bad boy

►

Koro

20:02

Problem: Compute the area of the region cut from the plane $x+y+z = a$ by the cylinder $x^2 +y^2 = a^2$.
Since I'm confused at parametrizing the region, here's a solution that I came up with:
$z=a-x-y$ so $z_x=-1=z_y$
The required area $a(S)= \int\int_T \sqrt{1+z_x^2+z_y^2}dxdy$.
What is $T$? $T$ is basically the projection of the plane on the cylinder. So $T$ is the region: {(x,y):x^2+y^2<=a}.
Using polar coordinates: $a(S)=\int_{0}^{2\pi}\int_{r=0}^a\sqrt 3 r dr d\theta=\sqrt 3\pi a^2$
Is that correct? Thanks.

Vrouvrou

20:16

hello, what is the idea to prove that $f(x,y)=x^y^4$ is lipshitzien over y on $\{(x,y), |x|\leq1,|y|\leq 3\}$

Khallil

$f(x,y) = (x^{y})^{4}$?

Vrouvrou

no sorry $f(x,y)=x^2+y^4$

Khallil

Are there equivalent conditions for being Lipschitz?

I vaguely recall there being something about a bounded first derivative

Vrouvrou

i don't know

Khallil

Also is this Lipschitz in $x$ or $y$?

Vrouvrou

20:27

in y

Khallil

According to Wikipedia
| We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line: Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous

Looks like a shortcut?

Ted Shifrin

@copper Yikes!

It's easy to prove with the Mean Value Theorem that any $C^1$ function is Lipschitz on a compact connected subset.

copper.hat

20:43

@TedShifrin Scarily early start to fire season!

leslie townes

ugh, awful. fire season is just every day now.

weirdly i think the bixby bridge is a different family of early-out-west folks than the bixbys who have so much named after them in long beach.

growing up in the west means growing up knowing somebody who has the last name of the street or neighborhood they live in. people ran out of ideas pretty quickly.

copper.hat

albany used to be named ocean view. apparently changed because there were so many ocean views that mail was misdelivered frequently. mild irony in choosing the name albany.

Ted Shifrin

@leslietownes I loved Bill Bixby!

leslie townes

20:59

he had to have been from one of the two families.

copper.hat

el cerrito used to benamed rust

leslie townes

people of my generation, but maybe not yours or anyone here, would know bixby from the movie ferris bueller's day off. in which ferris lived in a house in bixby knolls.

in the script he lived near chicago but the house is in long beach.

copper.hat

i think of the hulk

leslie townes

the ang lee hulk was filmed in berkeley.

copper.hat

i like ferrigno, he comes across as a decent human being

leslie townes

21:03

and of course the graduate.

copper.hat

the graduate said stanford, so when i watched it i was surprised to see telegraph avenue

leslie townes

yeah, durant and college is prominent too.

they also drive the wrong way over the bridge.

in "la la land" there is a scene with emma stone filmed a few blocks from my old house. i remember thinking "that's a cute apartment building" prior to the movie. maybe i have a second career as a location scout.

third career i guess.

copper.hat

educating rita was supposed to be in a uk university but was filmed in trinity, dublin. they painted all the mail boxes red (instead of the irish green). was a bit odd watching it and slowly realising that it was dublin (my birthdplace).

leslie townes

in one of the thin man movies nick and nora drive over the newly constructed bay bridge and you can see the hill in albany and they visit the race track. i think they visit the long beach airport (whose front facade is unchanged) in the same film.

worth a look.

Ted Shifrin

@leslietownes yes, the car goes into Lake Michigan, allegedly.

leslie townes

21:09

well if anyone's in the area i can drive us over to the bueller house. and the la la land apartment building.

copper.hat

my intermittent working on abstract algebra is not consistent with good progress. however, i realised that i am making mild movement when i encountered some $x^2+y^2=z^2$ problem and the first thing i did was take $\pmod{4}$.

Ted Shifrin

Mazltov!

copper.hat

must be the hafner winery trip yesterday.

Ted Shifrin

All my yelling at Under paid off.

copper.hat

a bit depressing, i have realised how barren my social life has become

:-)

Ted Shifrin

21:11

Social life?

copper.hat

exactly

leslie townes

my wife and i were married in a spot that dw griffith used for filming one of his non racist movies. there are scenes literally where we were married. i didn't know this until later, i am something of a film nerd and i was watching it and then it's like "wait isn't that where we cut the cake?" it was.

copper.hat

we got married at the brazilan room on tilden, all the irish folks ended up getting sun burned.

leslie townes

my social life is texting. i had dinner with one of my work friends a few weeks ago. but only one time in a year. no general socializing.

copper.hat

my sort of interaction style doesn't really work remotely, unfortunately. people either get mildly offended or think wtf

leslie townes

21:16

i'm on the wtf side, not mildly offended.

copper.hat

besides, there is a huge difference between chatting with a cute girl on the train and texting some girl.

Clarinetist

Good day all -

Ted Shifrin

Your wife will be glad to hear.

Hi Clarinet.

leslie townes

yeah, basically still on the wtf side.

Clarinetist

I suppose you've probably heard about Nature moving to open access and Nature Neuroscience opening it up with a $11,528 USD open-access fee?

leslie townes

21:18

clarinetist, i had not, although there is something mildly hilarious about a $10k+ "open access" fee.

copper.hat

@TedShifrin i chat with anyone who will listen. my wife knows i am a loyalty freak.

Clarinetist

Here you go: nature.com/articles/…

> Beginning in January 2021, we and our sister journals, including Nature, became ‘transformative’ journals. This means that authors of accepted primary research articles can now choose to have their papers published under a ‘gold’ open access model... the cost of publication is covered by an Article Processing Charge (APC)... The APC for Nature Neuroscience in 2022 is €9,500/US $11,390/£8,290

I think the additional $138 USD comes from something else, can't remember

copper.hat

does that mean to publish a single paper you need to pay $11k?

leslie townes

journals don't do a thing except provide their names. i am no longer an academic so i don't care, but if i had professional security, i would not review for any journal that bled libraries dry. i understand that people without professional security have to do about anything.

Clarinetist

@copper.hat That is how I am interpreting that, yes, for Open Access

copper.hat

21:22

god, i wasted so much of my graduate time doing detailed reviews

i would read references, etc

how stupid i am

Clarinetist

I've only done one journal review in my lifetime. Got about $240 for it and only spent 2 hours.

leslie townes

i did a lot of reviewing on behalf of my advisor. none of it felt exploitative but it was unpaid work and i don't do unpaid work anymore.

copper.hat

well, i have great respect for my advisor, but he certainly plumbed me for value.

so many stories, not really to be shared here

leslie townes

maybe over a pint at ben and nicks.

copper.hat

he is a concentration camp survivor.

any time you want :-)

i wonder if i listen to johnny nash's "i can see clearly now" enough times if it will improve the mood...

leslie townes

21:25

i'd need a dirty big glass of porter, 8% minimum.

copper.hat

:-) i usually stick with their plonk.

leslie townes

the journaling industry does depend in large part upon people feeling that they have to do stuff.

copper.hat

being raised catholic certainly helps

Ted Shifrin

Got $240? Every journal article I had to review was part of my professional obligation. When I've reviewed books pre-publication for publishers, I've usually gotten a few books out of it.

copper.hat

i never got paid for reviews

wth???

Ted Shifrin

21:26

Yeah, I've never heard of such a thing.

copper.hat

oligopolies. the only way to do business.

leslie townes

i was also never paid for anything. sometimes the publisher would send a book gratis and then ask for a review, which felt a little bit like pressure, but i ignored it.

Clarinetist

@copper.hat Yeah, one of the former faculty I used to work with is the associate editor of a journal and was looking for a statistical reviewer to settle the debate between two reviewers about the appropriateness of a statistical method.

copper.hat

cool!

i really am in a truly foul mood.

Ted Shifrin

A number of my editor friends knew they could count on me to be tough and reject stuff, so I got mostly things I rejected. One time I was sent an article by one of my teachers and someone I admire, so I was very flattered, but didn't have the time to do it justice.

calls referee on copper

copper.hat

21:28

:-)

leslie townes

the people who advertise textbooks have utterly no connection to the subject matter. they would visit and it would be like, here are three or four recent graduates of charm school here to bother you.

copper.hat

life would be so much simpler if people (including myself) were up front

leslie townes

this is how they sold opiate pills everywhere. you get young people to bother people relevant to the delivery of the product, and just push, push, push.

maybe i'm cynical. in my business if you make a guess at the worst reason why something is happening it is usually the right one.

Ted Shifrin

"Maybe" you're cynical?

copper.hat

unfortunately i think that is true

leslie townes

21:37

:D

copper.hat

i mean @leslietownes'sd comment

although, i have to say, that when i was at my most successful it was because i trusted people to do what they said they would do

Ted Shifrin

When I was associate department head for 8 years, I did lots of things I should have assigned to a secretary to do, but I figured I was more efficient and more likely to do them correctly the first time.

leslie townes

same here. formally i am assigned a secretary but any request is weighted by, would this actually be faster, and usually the answer is no.

it can take a whole lot of time to tell someone how you want something to be done in that time you can just do it.

copper.hat

there was line in What They Don't Teach You at Harvard Business School that took me years to understand, "spend 5 hours to save 5 minutes"

3

leslie townes

we have this all the time with huge corporate clients. some policy somewhere will say, you need to do X via Y form. ok, filling out the form will take 5 hours, and in 5 minutes i can tell you what you are not going to like if i filled out the form.

copper.hat

21:45

:-)

leslie townes

spoiler alert, we fill out the form.

Ted Shifrin

And charge $400/hr to do so.

leslie townes

it would be ungentlemanly to do otherwise.

Ted Shifrin

But of course.

Yet another interview to do in a few. It's a good thing I'm getting towards the end. After a dozen or so I'm sorta sick of it ... although the last few have been quite fun.

copper.hat

22:06

strange. a 2 word text from my son changed the sign on mood. becoming emo as i age

Ted Shifrin

23:30

@copper.hat $100 prize for guessing the two words.

copper.hat

:-)

Odestheory12

I have to compute the following limit $$\lim_{n \to{\infty}} \begin{bmatrix} 1+x & -x & 0 & 0 & \cdots & 0 & 0 \\ \frac{-1}{2} & 1+\frac{x}{2} & -x & 0 & \cdots & 0 & 0 \\ 0 & \frac{-1}{3} & 1+\frac{x}{3} & -x & \cdots & 0 & 0 \\ 0 & 0 & \frac{-1}{4} & 1+\frac{x}{4} & \cdots & 0 & 0 \\ \vdots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 1+ \frac{x}{n-1} & -x\\ \\ 0 & 0 & 0 & 0 & \cdots & \frac{-1}{n} & 1+ \frac{x}{n}\\ \end{bmatrix}$$

Ted Shifrin

In what world are you computing that limit?

Odestheory12

I can see the limit is gonna be $e^x$, and with some computations I got that $A_n=\left(1+\displaystyle\frac{x}{n}\right)A_{n-1}-\left(\displaystyle\frac{x}{n}\right)A_{n-2}$

Ted Shifrin

Huh?

Are you taking a limit of the determinant or something?

Odestheory12

23:36

Yeah sorry

Ted Shifrin

So $A_n$ is the determinant of the $n\times n$ matrix?

Odestheory12

Yeah

copper.hat

The determinant is continuous.

Odestheory12

So I want to prove by induction that $\displaystyle A_n = \sum_{i=0}^{n}\dfrac{x^i}{i!}$ with the recurrence I found

Ted Shifrin

Not when you're varying the sizes of the matrices, @copper.

copper.hat

23:37

Ahh, I did not see that.

Ted Shifrin

No det in that sentence, @Odes.

Odestheory12

I've seen the cases $A_1$, $A_2$ and such, but meh I can't see how to proceed.

Ted Shifrin

So you need to say $A_{-1}=0$ and $A_0=1$, I guess.

Odestheory12

I noticed that $A_n = \frac{x^n}{n!}+A_{n-1}$ but not sure how to get this from my recurrence relation

Ted Shifrin

Well, does induction do it?

Odestheory12

23:40

Uhm, not sure how to proceed. What I am trying to do is:

Since I noticed $A_n = \frac{x^n}{n!}+A_{n-1}$, then the first recurrence relation $A_n=\left(1+\displaystyle\frac{x}{n}\right)A_{n-1}-\left(\displaystyle\frac{x}{n}\right)A_{n-2}$ could be rewritten somehow

I tried to see if its "telescoping" but no luck :(

Ted Shifrin

No, you're confusing things. You didn't notice $A_n$ is that formula. You want to prove that.

Odestheory12

Right

Well, that's my problem, I don't know how to use $A_n=\left(1+\displaystyle\frac{x}{n}\right)A_{n-1}-\left(\displaystyle\frac{x}{n}\right)A_{n-2}$ to prove that

I tried to compute $A_{n-1}$ and substitute it in that relation but meh

Ted Shifrin

Do you know how to do proofs by induction?

Odestheory12

Yeah, I know I want to compute $A_{n+1}$

But should be the same than doing the $n-1$ case implies $n$ case, shouldn't it?

Ted Shifrin

Assuming your conjectured formula is correct for $A_{n-2}$ and $A_{n-1}$, plug those in and do the algebra in your recursion formula and hope it gives you your formula for $A_n$.

It does work, and it's not too hard.

Odestheory12

23:47

Do you mean to plug it here? $A_n = \frac{x^n}{n!}+A_{n-1}$

Ted Shifrin

No.

You conjectured that $A_n = \sum_{i=0}^n x^i/i!$.

Odestheory12

Yeah

Ted Shifrin

Prove that by (complete) induction as I indicated.

Odestheory12

But I don't see how to get a sum from my recurrence relation

Ted Shifrin

Did you read what I wrote?

Odestheory12

23:49

Yeah but I guess I didn't understand it. Let me re-read it and think.

Ted Shifrin

Put in the conjectured formula for $n-1$ and $n-2$ into your recursion formula.

Odestheory12

Ah!!

God. I was trying to derive the sumation from my recurrence going backwards until $A_1$.

Ted Shifrin

Induction is a powerful technique.

Odestheory12

Hoping to see some kind of "simplification".

Wrong latex line :D

$\displaystyle A_n=\left(1+\displaystyle\frac{x}{n}\right)\sum_{i=0}^{n-1}\dfrac{x^i}{i!}-\left(\displaystyle\frac{x}{n}\right)\sum_{i=0}^{n-2}\dfrac{x^i}{i!}$

You meant that right :)

Ted Shifrin

Yes!

Odestheory12

23:53

And yes now its telescoping

Thanks :)

Ted Shifrin

Do the algebra intelligently and economically. You don't even have to write out the telescoping. Just cancel.

You're welcome.

Odestheory12

Yeah the entire sumation cancels after introducing the fraction into the sum.

Ted Shifrin

You don't even need to do that!

Just look at $\sum_{i=0}^{n-1} = \sum_{i=0}^{n-2} + (n-1)\text{-term}$.

Odestheory12

Ah!! right, thanks :)

I guess what I was doing is stupid. I was trying to rewrite the recursion formula $A_n=\left(1+\displaystyle\frac{x}{n}\right)A_{n-1}-\left(\displaystyle\frac{x}{n}\right)A_{n-2}$ in the $A_n = \frac{x^n}{n!}+A_{n-1}$ form.

Ted Shifrin

Well, that's what it turns out to be, so that's fine.

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$Mathematics$ Mathematics